Substance

he Domino Method of General Mixed-Integer Nonlinear Programming Applied to a Speed-Reducer Design with Discrete Variables

Jsun Yui Wong

The following computer program seeks to solve Example 13 of Jaberipour and Khorram [22, p. 3329] with one modification.  The modification is that the discrete values of Rao and Xiong [45, pp. 1109-1110] are used in the following computer program.

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
107 A(1)=2.6+    +FIX (RND*11)  *.1
109 A(2)=.7+   FIX  (    RND*2   ) *.1
111 A(3)=17+FIX(  RND*12)
113 A(4)=7.3+   FIX(  RND *11)  *.1
114 A(5)=7.3+   FIX(  RND *11)  *.1
115 A(6)=2.9+   FIX(RND*101      )  *.01
116 A(7)=5+     FIX(RND*51     )    *.01
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 7
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*7)
144       IF JJJJ<-31995 THEN  162 ELSE 145
145       IF B=3 GOTO 162
147 IF B=1 OR B=2 OR B=4 OR B=5 GOTO 150 ELSE IF B=6 OR B=7 GOTO 158
150 IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2)*.1                                     ELSE X(B)=A(B)+FIX(1+RND*2)*.1
152 GOTO 179
158 IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2)*.01                                     ELSE X(B)=A(B)+FIX(1+RND*2)*.01
161 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 7
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
181 IF ABS(X(2))<.0000001 THEN 1670
182    X(8)=    12    -X(1)/X(2)
185    X(9)= -5  +X(1)/X(2)
188    X(10)=    40    -X(2)*X(3)
190 IF ABS(  X(4)) <.0000001 THEN 1670
191    X(11)= 1-      (  1.5*X(6)   +1.9 )        /X(4)
192 IF ABS(  X(5)) <.0000001 THEN 1670
193    X(12)= 1-      (  1.1*X(7)   +1.9 )        /X(5)
194 IF   ABS((   X(1)  *  X(2)^2  *    X(3)        )   )<.0000001 THEN 1670
195 X(13)=      1-   27/(   X(1)  *  X(2)^2  *    X(3)        )
196 IF ABS ((   X(1)  *  X(2)^2  *    X(3) ^2     )    )<.0000001 THEN 1670
197 X(14)=      1- 397.5 /(   X(1)  *  X(2)^2  *    X(3) ^2     )
198 IF ABS ( (   X(2)  *  X(3)  *    X(6) ^4     )    )<.0000001 THEN 1670
199 X(15)=      1-     1.93*X(4)^3     / (   X(2)  *  X(3)  *    X(6) ^4     )
200 IF ABS( (   X(2)  *  X(3)  *    X(7)  ^4    )       )<.0000001 THEN 1670
201 X(16)=      1-   1.93*X(5)^3 / (   X(2)  *  X(3)  *    X(7)  ^4    )
204 IF ABS( (110*X(6)^3   )  )   <.0000001 THEN 1670
205 X(17)=   1- (     (  745*X(4)/(X(2)*X(3))   )^2    +    16900000#)  ^.5             /     (110*X(6)^3   )
208 IF  ABS(   (85*X(7)^3   )       )<.0000001 THEN 1670
209 X(18)=   1- (     (  745*X(5)/(X(2)*X(3))   )^2    +    157500000#)  ^.5             /     (85*X(7)^3   )
221 IF X(1)<2.6 THEN 1670
222 IF X(2)<.7 THEN 1670
223 IF X(3)<17 THEN 1670
224 IF X(4)<7.3 THEN 1670
225 IF X(5)<7.3 THEN 1670
226 IF X(6)<2.9 THEN 1670
227 IF X(7)<5 THEN 1670
231 IF X(1)>3.6 THEN 1670
232 IF X(2)>.8 THEN 1670
233 IF X(3)>28 THEN 1670
234 IF X(4)>8.3 THEN 1670
235 IF X(5)>8.3 THEN 1670
236 IF X(6)>3.9 THEN 1670
237 IF X(7)>5.5 THEN 1670
421 FOR J44=8 TO 18
422 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
423 NEXT J44
425 PTOTAL=P(8)+P(9)+P(10) +P(11)+P(12)+P(13)+P(14)  +P(15)+P(16)+P(17)+P(18)
569  P=      -.7854*X(1)*X(2)^2   *(3.3333*X(3)^2+14.9334*X(3)-43.0934   )           +1.508*X(1)*(X(6)  ^2 +X(7)^2  )  -7.4777*(X(6)^3 +X(7)  ^3  )  -.7854*(    X(4)*X(6)^2   +X(5)*X(7)  ^2  )    +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 18
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-3001 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),M,JJJJ
1990 PRINT A(8),A(9)
1991 PRINT A(10),A(11),A(12)
1992 PRINT A(13),A(14),A(15)
1993 PRINT A(16),A(17),A(18)
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The complete output through JJJJ=-31954 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

3.500000007450581       .6999999880790711         17
7.300000190734863       7.800000384449959         3.360000202432275
5.290000209584832      -3000.959888639683        -31962
6.999999904206819       9.579318072105991D-08
28.10000020265579       4.931505500304229D-02     1.038462149137188D-02 
7.391525082700623D-02   .1979985015331917         .5049811426052296
.9017185718852978       8.711664113408416D-03     1.879905546478974D-03

3.500000007450581       .6999999880790711       17
7.300000369548798       7.800000004470348       3.360000038519502
5.290000129491091      -3000.95978886027       -31954
6.999999904206819       9.579318072105991D-08
28.10000020265579       .0493151119708305       1.038458457719901D-02 
7.391525082700623D-02   .1979985015331917       .5049810096335322
.9017185802965785       8.711518741241028D-03   1.879860283591869D-03

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was 15 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
 
References

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[43] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge University Press, 2000.

[44] W. L. Price, Global Optimization by Controlled Random Search.  Journal of Optimization Theory and Applications, July 1983, Volume 40, Issue 3, pp. 333-348.

[45] Singiresu S. Rao, Ying Xiong (2005), A Hybrid Genetic Algorithm for Mixed-Discrete Design Optimization.  Transaction of the ASME, Vol. 127, November 2005, pp. 1100-1112.

[46] John R. Rice, Numerical Methods, Software, and Analysis, Second Edition.  Academic Press, 1993.

[47] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

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[50] Jung-Fa Tsai (2010), Global Optimization for Signomial Discrete Programming Problems in Engineering Design.  Engineering Optimization, 42:9, pp. 833-843.

[51] L.-W. Tsai, A. P. Morgan, Solving the Kinematics of the Most General Six- and Five-Degree-of Freedom Manipulators by Continuation Methods.  Journal of Mechanisms, Transmissions, and Automation in Design, June 1985, Vol. 107, pp. 189-200.

[52] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[53] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[54] Jsun Yui Wong (2011, May 15).  The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/

[55] Jsun Yui Wong (2011, May 5).  The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/

[56] Jsun Yui Wong (2011, July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[57] Jsun Yui Wong (2012, April 10).  The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design.  Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/

[58] Jsun Yui Wong (2012, April 21).  The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/

[59] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[60] Jsun Yui Wong (2013, January 14).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition.   Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/

[61] Jsun Yui Wong (2013, January 24).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables.   Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/

The Domino Method of General Mixed-Integer Nonlinear Programming Applied to Weight Minimization of a Speed Reducer

Jsun Yui Wong

The following computer program seeks to solve Example 13 of Jaberipour and Khorram [22, p. 3329].  The source of this example is Li and Papalambros [34].

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
107 A(1)=2.6+RND
109 A(2)=.7+(RND*.1)
111 A(3)=17+RND*11
113 A(4)=7.3+RND
114 A(5)=7.3+RND
115 A(6)=2.9+RND
116 A(7)=.5+(RND*5)
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 7
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*7)
144       IF JJJJ<-31995 THEN  162 ELSE 164
145       IF B=3 OR B=4 GOTO 164
147       IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625                                     ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625                                    ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 7
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
181 IF ABS(X(2))<.0000001 THEN 1670
182    X(8)=    12    -X(1)/X(2)
185    X(9)= -5  +X(1)/X(2)
188    X(10)=    40    -X(2)*X(3)
190 IF ABS(  X(4)) <.0000001 THEN 1670
191    X(11)= 1-      (  1.5*X(6)   +1.9 )        /X(4)
192 IF ABS(  X(5)) <.0000001 THEN 1670
193    X(12)= 1-      (  1.1*X(7)   +1.9 )        /X(5)
194 IF   ABS((   X(1)  *  X(2)^2  *    X(3)        )   )<.0000001 THEN 1670
195 X(13)=      1-   27/(   X(1)  *  X(2)^2  *    X(3)        )
196 IF ABS ((   X(1)  *  X(2)^2  *    X(3) ^2     )    )<.0000001 THEN 1670
197 X(14)=      1- 397.5 /(   X(1)  *  X(2)^2  *    X(3) ^2     )
198 IF ABS ( (   X(2)  *  X(3)  *    X(6) ^4     )    )<.0000001 THEN 1670
199 X(15)=      1-     1.93*X(4)^3     / (   X(2)  *  X(3)  *    X(6) ^4     )
200 IF ABS( (   X(2)  *  X(3)  *    X(7)  ^4    )       )<.0000001 THEN 1670
201 X(16)=      1-   1.93*X(5)^3 / (   X(2)  *  X(3)  *    X(7)  ^4    )
204 IF ABS( (110*X(6)^3   )  )   <.0000001 THEN 1670
205 X(17)=   1- (     (  745*X(4)/(X(2)*X(3))   )^2    +    16900000#)  ^.5             /     (110*X(6)^3   )
208 IF  ABS(   (85*X(7)^3   )       )<.0000001 THEN 1670
209 X(18)=   1- (     (  745*X(5)/(X(2)*X(3))   )^2    +    157500000#)  ^.5             /     (85*X(7)^3   )
221 IF X(1)<2.6 THEN 1670
222 IF X(2)<.7 THEN 1670
223 IF X(3)<17 THEN 1670
224 IF X(4)<7.3 THEN 1670
225 IF X(5)<7.3 THEN 1670
226 IF X(6)<2.9 THEN 1670
227 IF X(7)<.5 THEN 1670
231 IF X(1)>3.6 THEN 1670
232 IF X(2)>.8 THEN 1670
233 IF X(3)>28 THEN 1670
234 IF X(4)>8.3 THEN 1670
235 IF X(5)>8.3 THEN 1670
236 IF X(6)>3.9 THEN 1670
237 IF X(7)>5.5 THEN 1670
421 FOR J44=8 TO 18
422 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
423 NEXT J44
425 PTOTAL=P(8)+P(9)+P(10) +P(11)+P(12)+P(13)+P(14)  +P(15)+P(16)+P(17)+P(18)
569  P=      -.7854*X(1)*X(2)^2   *(3.3333*X(3)^2+14.9334*X(3)-43.0934   )           +1.508*X(1)*(X(6)  ^2 +X(7)^2  )  -7.4777*(X(6)^3 +X(7)  ^3  )  -.7854*(    X(4)*X(6)^2   +X(5)*X(7)  ^2  )    +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 18
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-2994.48 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),M,JJJJ
1990 PRINT A(8),A(9)
1991 PRINT A(10),A(11),A(12)
1992 PRINT A(13),A(14),A(15)
1993 PRINT A(16),A(17),A(18)
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The complete output through JJJJ=-31992 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

3.499999946131225       .6999999880987415       17.00000000119371
7.300000190937412       7.71532001879137        3.350214666815489
5.286654465101151      -2994.471024338103      -31995
6.999999991946404       8.053596234347538D-09
28.1000002014858        5.132578147342792D-02   6.452482576024821D-10
7.391523353624762D-02   .1979984876399762       .4991722143650292
.9046439015628293       1.178675768098714D-10   2.300630519425084D-11

3.499999956249128       .6999999896476371       17.00000010657121
7.3000002283353         7.715320031828895       3.350214674428932
5.286654470357651      -2994.47105941194       -31994
6.999999988555797       1.144420358478726D-08
28.10000010139032       .0513257847690784       1.585633777057005D-09
7.391524723521015D-02   .1979985034503124       .4991722154330369
.9046439022607501       6.975583249912987D-09   3.015836691488971D-09

3.499999940666056       .699999988090363        17.00000000643012
7.300000192317069       7.715320028459313       3.350214666869112
5.286654465346311      -2994.471023452137      -31992
6.999999999693941       3.060591780013056D-10
28.10000019796275       5.132578164170333D-02   1.86337890950572D-09
7.391523353627415D-02   .1979984868625425       .4991722142614059
.9046439012502803       1.671921628920714D-10   1.607105698520073D-10

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was 32 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
 
References

[1] J. S. Arora, Introduction to Optimun Design.  New York: McGraw-Hill; 1989.

[2] Dana H. Ballard, C. O Jelinek, R. Schinzinger (1974), An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Volume 17, Number 3, pp. 261-266.

[3] M. C. Bartholomew-Biggs, A Numerical Comparison between Two Approaches to the Nonlinear Programming Problem, in Towards Global Optimization 2, edited by L. C. W. Dixon, G. P. Szego, pp. 293-312.  North-Holland Publishing Company 1978.

[4] J. Bracken, G. P. McCormick, Selected Applications of Nonlinear Programming.  New York: John Wiley and Sons, 1968.

[5] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition.  Brooks/Cole 2011.

[6] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.

[7] C. A. C. Coello, E. M. Mesura (2002), Constraint Handling in Genetic Algoritmms through the Use of Dominance-Based Tournament Selection.  Advanced Engineering Informatics, 16 (3), 193-203.

[8] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[9] Kalyanmoy Deb (2000), An Efficient Constraint Handling Method for Genetic Algorithms.  Computer Methods in Applied Mechanics and Engineering, 186 (2000), 311-338.

[10] R. J. Duffin, E. L. Peterson, C. Zener, Geometric Programming Theory and Applications.  John Wiley and Sons, 1967.

[11] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs.  Mathematical Programming, Vol. 36:307-339, 1986.

[12] F. Erbatur,  O. Hasancebi, I. Tutuncu, H. Kilic (2000) Optimal Design of Planar and Spcae Structures with Genetic Algorithms.  Computers and Structures, 75 (2), 209-224.

[13] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[14] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[15] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[16] Christodoulos A. Floudas, Nonlinear and Mixed-Integer Optimization.  Oxford University Press, 1995.

[17] C. A. Floudas, Deterministic Global Optimization.  Kluwer Academic Publishers, 2000.

[18] S. Gholizadeh, A. Barzegar (2012), Shape Optimization of Structures for Frequency Constraints by Sequential Harmony Search Algorithm, Engineering Optimization, DOI:10.1080/0305215X.2012.704028.

[19] Amos Gilat and Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB.  John Wiley and Sons, Inc. 2008.

[20] Donald Greenspan, Vincenzo Casulli, Numerical Analysis for Applied Mathematics, Science, and Enginerring.  Addison-Wesley Publishing Company, 1988.

[21] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Springer-Verlag 1981.

[22] Majid Jaberipour, Esmaile Khorram (2010), Two Improved Harmony Search Algorithms for Solving Engineering Optimization Problems.  Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3316-3331.

[23] M. Jaberipour, E. Khorram (2010), Solving the Sum-of-Ratios Problems by a Harmony Search Algorithm.  Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[24] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed Discrete Engineering Optimization Problems.  Engineering Optimization, 05/2011, 43:507-523.

[25] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[26] B. K. Kannen , S. N. Kramer (1994), An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design.  Journal of Mechanical Design, 116, pp. 405-411.

[27] R. Baker Kearfott, Some Tests of Generalized Bisection.  ACM Transactions on Mathematical Software, Vol.13, No. 3, September 1987, pages 197-220.

[28] Esmaile Khorram, Majid Jaberipour, Harmony Search Algorithm for Solving Combined Heat and Power Economic Dispatch Problems.  Energy Conversion and Management 52 (2011) 1550-1554.

[29] G. R. Kocis, I. E. Grossmann, Relaxation Strategy for the Structural Optimization of Process Flow Sheets.  Ind. Eng. Chem. Res., 26 (9):1869 (1987).

[30} Sonia Krzyworzcka, Extension of the Lanczos and CGS Methods to Systems of Nonlinear Equations.  Journal of Computational and Aplied Mathematics 69 (1996) 181-190.

[31] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[32] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[33] K. S. Lee, Z. W. Geem (2005), A New Meta-Heuristic Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice.  Computer Methods in Applied Mechanics and Engineering 194 (2005) 3902-3933.

[34] H. L. Li, P. Papalambros (1985), A Production System for Use of Global Optimization Knowledge.  Journal of Mechanisms, Transmissions, and Automation in Design.  June 1985, Vol. 107, 277-284.

[35] Ya-Zhang Luo, Guo-Jing Tang, Li-Ni Zhou, Hybrid Approach for Solving Systems of Nonlinear Equations Using Chaos Optimization and Quasi-Newton Method.  Applied Soft Computing 8 (2008) 1068-1063.

[36] M. Mahdavi, M. Fesanghary, E. Damangir (2007), An Improved Harmony search Algorithm for Solving Optimization Problems.  Applied Mathematics and Computation, 188 (2), 1567-1579.

[37] C. D. Maranas, C. A. Floudas, Finding All Solutions of Nonlinearly Constrained Systems of Equations.  Journal of Global Optimization, 7(2):143-182, 1995.

[38] Harry M. Markowitz and Alan S. Mann.  On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[39] Keith Meintjes and Alexander P. Morgan, Chemical Equilibrium Systems as Numerical Test Problems.  ACM Transactions on Mathematical Software, Vol. 16, No. 2, June 1990, Pages 143-151.

[40] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[41] Yuanbin Mo, Hetong Liu, Qin Wang,Conjugate Direction Particle Swarm Optimization Solving Systems of Nonlinear Equations. Computers and Mathematics with Applications 57 (2009) 1877-1882.

[42] Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, An Introduction to Interval Analysis.  Cambridge University Press, 2009.

[43] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge University Press, 2000.

[44] W. L. Price, Global Optimization by Controlled Random Search.  Journal of Optimization Theory and Applications, July 1983, Volume 40, Issue 3, pp. 333-348.

[45] Singiresu S. Rao, Ying Xiong (2005), A Hybrid Genetic Algorithm for Mixed-Discrete Design Optimization.  Transaction of the ASME, Vol. 127, November 2005, pp. 1100-1112.

[46] John R. Rice, Numerical Methods, Software, and Analysis, Second Edition.  Academic Press, 1993.

[47] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[48] E. Sandgren (1990), Nonlinear Integer and Discrete Programming in Mechanical Design Optimization.  Journal of Mechanical Design, 112, pp. 223-229.

[49] R. Schinzinger (1965), Optimization in Electromagnetic System Design, in Recent Advances in Optimization Techniques, edited by A. Lavi and T. P. Vogel, John Wiley and Sons, pp. 163-213.

[50] Jung-Fa Tsai (2010), Global Optimization for Signomial Discrete Programming Problems in Engineering Design.  Engineering Optimization, 42:9, pp. 833-843.

[51] L.-W. Tsai, A. P. Morgan, Solving the Kinematics of the Most General Six- and Five-Degree-of Freedom Manipulators by Continuation Methods.  Journal of Mechanisms, Transmissions, and Automation in Design, June 1985, Vol. 107, pp. 189-200.

[52] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[53] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[54] Jsun Yui Wong (2011, May 15).  The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/

[55] Jsun Yui Wong (2011, May 5).  The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/

[56] Jsun Yui Wong (2011, July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[57] Jsun Yui Wong (2012, April 10).  The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design.  Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/

[58] Jsun Yui Wong (2012, April 21).  The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/

[59] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[60] Jsun Yui Wong (2013, January 14).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition.   Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/

[61] Jsun Yui Wong (2013, January 24).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables.   Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/

The Domino Method of General Mixed-Integer Nonlinear Programming Applied to a Tension/Compression String Problem

Jsun Yui Wong

The following computer program seeks to solve the problem considered by Arora [1], Coello [7], Mahdavi, Fesanghary, and Damangir [35], and Jaberipour and Khorram [22, p. 3326], among others.  Table 10 in Jaberipour and Khorram [22, p. 3327] is an informative summary.

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
107 A(1)=.05+RND*1.95
109 A(2)=.25+RND*1.05
111 A(3)=2+RND*13
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 3
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*3)
144       IF JJJJ<-31995 THEN  162 ELSE 164
145       IF B=3 OR B=4 GOTO 164
147       IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625                                     ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625                                    ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 4
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
184    X(4)=     1      - ( X(1)       +X(2)  )/1.5
187 IF   (  X(2)^2   *X(3)   )   <.0000001 THEN 1670
188    X(5)=  -  1      + (    140.45*X(1)     )/    (  X(2)^2   *X(3)   )
191 IF    (71785!*  X(1)^4    )    <.0000001 THEN 1670
192    X(6)=  -  1      + (   X(2)^3  *X(3)     )/    (71785!*  X(1)^4    )
193  IF   (5108*  X(1)^2    )  <.0000001 THEN 1670
194  IF   (12566*    (    X(2)*X(1)^3    -X(1)^4   )  )   <.0000001 THEN 1670
195    X(7)=    1      - 1   /    (5108*  X(1)^2    )                               -( 4*X(2)^2 - X(2)*X(1)    )/(12566*    (    X(2)*X(1)^3    -X(1)^4   )  )
212 IF X(1)<.05 THEN 1670
215 IF X(1)>2 THEN 1670
217 IF X(2)<.25 THEN 1670
218 IF X(2)>1.3 THEN 1670
219 IF X(3)<2 THEN 1670
220 IF X(3)>15 THEN 1670
221 FOR J44=4 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569  P=      -(X(3)+2  )*X(2)*X(1)^2    +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.0127 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The output through JJJJ=-31520 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

5.071918001974184D-02    .3338299604682986        12.76869949893942
.7436339063413064        4.006067133769477        5.053013563127706D-12
3.203270981799733D-12   -1.268270854522712D-02   -31880

5.153511251424495D-02    .3530254249515273        11.50876349825394
.7302929750228185        4.046430868310082        2.157382605894043D-11
8.880257640342393D-13   -1.266566625514161D-02   -31647

5.260890402985393D-02    .3792528346950238        10.08054971889733
.7120921741834149        4.096113611344759        4.206801573758412D-12
7.677275482009804D-12   -1.268043177161621D-02   -31569

5.164043744300604D-02    .3555491093087384        11.35780925408395
.7285403021655037        4.051471252459476        2.604265886274604D-09
8.995355849084063D-11   -1.266527597819639D-02   -31520

The other candidate solutions have objective function values of -1.269936784221014D-02
at JJJJ=31809, -1.269810946101729 at JJJJ=-31548, and -1.269633438853511D-02 at JJJJ=-31525.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was 29 minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
 
References

[1] J. S. Arora, Introduction to Optimun Design.  New York: McGraw-Hill; 1989.

[2] Dana H. Ballard, C. O Jelinek, R. Schinzinger (1974), An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Volume 17, Number 3, pp. 261-266.

[3] M. C. Bartholomew-Biggs, A Numerical Comparison between Two Approaches to the Nonlinear Programming Problem, in Towards Global Optimization 2, edited by L. C. W. Dixon, G. P. Szego, pp. 293-312.  North-Holland Publishing Company 1978.

[4] J. Bracken, G. P. McCormick, Selected Applications of Nonlinear Programming.  New York: John Wiley and Sons, 1968.

[5] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition.  Brooks/Cole 2011.

[6] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.

[7] C. A. C. Coello, E. M. Mesura (2002), Constraint Handling in Genetic Algoritmms through the Use of Dominance-Based Tournament Selection.  Advanced Engineering Informatics, 16 (3), 193-203.

[8] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[9] Kalyanmoy Deb (2000), An Efficient Constraint Handling Method for Genetic Algorithms.  Computer Methods in Applied Mechanics and Engineering, 186 (2000), 311-338.

[10] R. J. Duffin, E. L. Peterson, C. Zener, Geometric Programming Theory and Applications.  John Wiley and Sons, 1967.

[11] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs.  Mathematical Programming, Vol. 36:307-339, 1986.

[12] F. Erbatur,  O. Hasancebi, I. Tutuncu, H. Kilic (2000) Optimal Design of Planar and Spcae Structures with Genetic Algorithms.  Computers and Structures, 75 (2), 209-224.

[13] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[14] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[15] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[16] Christodoulos A. Floudas, Nonlinear and Mixed-Integer Optimization.  Oxford University Press, 1995.

[17] C. A. Floudas, Deterministic Global Optimization.  Kluwer Academic Publishers, 2000.

[18] S. Gholizadeh, A. Barzegar (2012), Shape Optimization of Structures for Frequency Constraints by Sequential Harmony Search Algorithm, Engineering Optimization, DOI:10.1080/0305215X.2012.704028.

[19] Amos Gilat and Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB.  John Wiley and Sons, Inc. 2008.

[20] Donald Greenspan, Vincenzo Casulli, Numerical Analysis for Applied Mathematics, Science, and Enginerring.  Addison-Wesley Publishing Company, 1988.

[21] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Springer-Verlag 1981.

[22] Majid Jaberipour, Esmaile Khorram (2010), Two Improved Harmony Search Algorithms for Solving Engineering Optimization Problems.  Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3316-3331.

[23] M. Jaberipour, E. Khorram (2010), Solving the Sum-of-Ratios Problems by a Harmony Search Algorithm.  Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[24] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed Discrete Engineering Optimization Problems.  Engineering Optimization, 05/2011, 43:507-523.

[25] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[26] B. K. Kannen , S. N. Kramer (1994), An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design.  Journal of Mechanical Design, 116, pp. 405-411.

[27] R. Baker Kearfott, Some Tests of Generalized Bisection.  ACM Transactions on Mathematical Software, Vol.13, No. 3, September 1987, pages 197-220.

[28] Esmaile Khorram, Majid Jaberipour, Harmony Search Algorithm for Solving Combined Heat and Power Economic Dispatch Problems.  Energy Conversion and Management 52 (2011) 1550-1554.

[29] G. R. Kocis, I. E. Grossmann, Relaxation Strategy for the Structural Optimization of Process Flow Sheets.  Ind. Eng. Chem. Res., 26 (9):1869 (1987).

[30} Sonia Krzyworzcka, Extension of the Lanczos and CGS Methods to Systems of Nonlinear Equations.  Journal of Computational and Aplied Mathematics 69 (1996) 181-190.

[31] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[32] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[33] K. S. Lee, Z. W. Geem (2005), A New Meta-Heuristic Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice.  Computer Methods in Applied Mechanics and Engineering 194 (2005) 3902-3933.

[34] Ya-Zhang Luo, Guo-Jing Tang, Li-Ni Zhou, Hybrid Approach for Solving Systems of Nonlinear Equations Using Chaos Optimization and Quasi-Newton Method.  Applied Soft Computing 8 (2008) 1068-1063.

[35] M. Mahdavi, M. Fesanghary, E. Damangir (2007), An Improved Harmony search Algorithm for Solving Optimization Problems.  Applied Mathematics and Computation, 188 (2), 1567-1579.

[35] C. D. Maranas, C. A. Floudas, Finding All Solutions of Nonlinearly Constrained Systems of Equations.  Journal of Global Optimization, 7(2):143-182, 1995.

[37] Harry M. Markowitz and Alan S. Mann.  On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[38] Keith Meintjes and Alexander P. Morgan, Chemical Equilibrium Systems as Numerical Test Problems.  ACM Transactions on Mathematical Software, Vol. 16, No. 2, June 1990, Pages 143-151.

[39] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[40] Yuanbin Mo, Hetong Liu, Qin Wang,Conjugate Direction Particle Swarm Optimization Solving Systems of Nonlinear Equations. Computers and Mathematics with Applications 57 (2009) 1877-1882.

[41] Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, An Introduction to Interval Analysis.  Cambridge University Press, 2009.

[42] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge University Press, 2000.

[43] W. L. Price, Global Optimization by Controlled Random Search.  Journal of Optimization Theory and Applications, July 1983, Volume 40, Issue 3, pp. 333-348.

[44] Singiresu S. Rao, Ying Xiong (2005), A Hybrid Genetic Algorithm for Mixed-Discrete Design Optimization.  Transaction of the ASME, Vol. 127, November 2005, pp. 1100-1112.

[45] John R. Rice, Numerical Methods, Software, and Analysis, Second Edition.  Academic Press, 1993.

[46] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[47] E. Sandgren (1990), Nonlinear Integer and Discrete Programming in Mechanical Design Optimization.  Journal of Mechanical Design, 112, pp. 223-229.

[48] R. Schinzinger (1965), Optimization in Electromagnetic System Design, in Recent Advances in Optimization Techniques, edited by A. Lavi and T. P. Vogel, John Wiley and Sons, pp. 163-213.

[49] Jung-Fa Tsai (2010), Global Optimization for Signomial Discrete Programming Problems in Engineering Design.  Engineering Optimization, 42:9, pp. 833-843.

[50] L.-W. Tsai, A. P. Morgan, Solving the Kinematics of the Most General Six- and Five-Degree-of Freedom Manipulators by Continuation Methods.  Journal of Mechanisms, Transmissions, and Automation in Design, June 1985, Vol. 107, pp. 189-200.

[51] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[52] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[53] Jsun Yui Wong (2011, May 15).  The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/

[54] Jsun Yui Wong (2011, May 5).  The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/

[55] Jsun Yui Wong (2011, July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[56] Jsun Yui Wong (2012, April 10).  The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design.  Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/

[57] Jsun Yui Wong (2012, April 21).  The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/

[58] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[59] Jsun Yui Wong (2013, January 14).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition.   Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/

[60] Jsun Yui Wong (2013, January 24).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables.   Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/

The Domino Method of General Mixed-Integer Nonlinear Programming Applied to a Nonlinear Programming Test Problem with Equality Constraints

Jsun Yui Wong

The following computer program seeks to solve test problem 7 in Deb [7, p. 331]. This problem has five continuous variables and three equality constraints.

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
101 FOR J44=1 TO 5
103 A(J44)=-2.3 + RND*4.6
106 NEXT J44
113 REM     A(4)=20+FIX(RND*41)
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*5)
144       IF JJJJ<-31995 THEN  154 ELSE 164
145 REM   IF B=3 OR B=4 GOTO 164
147 REM   IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625                                     ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625                                    ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 5
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
192 IF  ( -1   -X(1)^3       ) <.00001 THEN 1670
194    X(2)=  ( -1   -X(1)^3       ) ^.3333333333333333#
195 REM      X(6)=X(2)         -.00954*X(3)
197 IF  ABS (5*X(4))    <.00001 THEN 1670
199 X(5)=      (  X(2)*X(3)    )   /  (5*X(4))
202 X(6)=   -10+X(1)^2+X(2)^2+X(3)^2+X(4)^2+X(5)^2
211 FOR J44=1 TO 2
212 IF X(J44)<-2.3 THEN 1670
215 IF X(J44)>2.3 THEN 1670
216 NEXT J44
217 FOR J44=3 TO 5
218 IF X(J44)<-3.2 THEN 1670
219 IF X(J44)>3.2 THEN 1670
220 NEXT J44
221 FOR J44=6 TO 6
222 IF ABS(  X(J44)      )      >0 THEN P(J44)=-9000000!*ABS(X(J44)  )  ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(6)
567  IF   (X(1)*X(2)*X(3)    *X(4)*X(5) ) >80 THEN 1670
569  P=      -EXP(X(1)*X(2)*X(3)    *X(4)*X(5) )    +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 6
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.0544 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The complete output through JJJJ=-29733 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

-1.71190810658521         1.589642492833478    -1.836945333049172
-.7652926124686905        .763129373222997      2.456368441983159D-15
-5.396309331019588D-02   -31805

-1.689683861167147         1.56378283341442     1.877220815462657
 .7479667056878516         .784945549962439     5.051514762044462D-15
-5.435684077803866D-02    -31212

-1.710483798634002         1.587990318229139    -1.83847905458375
 .7992999171951749        -.7305110074803548     2.250699626671349D-13
-5.416114633260474D-02    -29733

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was 25 minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
 
References

[1] Dana H. Ballard, C. O Jelinek, R. Schinzinger (1974), An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Volume 17, Number 3, pp. 261-266.

[2] M. C. Bartholomew-Biggs, A Numerical Comparison between Two Approaches to the Nonlinear Programming Problem, in Towards Global Optimization 2, edited by L. C. W. Dixon, G. P. Szego, pp. 293-312.  North-Holland Publishing Company 1978.

[3] J. Bracken, G. P. McCormick, Selected Applications of Nonlinear Programming.  New York: John Wiley and Sons, 1968.

[4] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition.  Brooks/Cole 2011.

[5] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.

[6] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[7] Kalyanmoy Deb (2000), An Efficient Constraint Handling Method for Genetic Algorithms.  Computer Methods in Applied Mechanics and Engineering, 186 (2000), 311-338.

[8] R. J. Duffin, E. L. Peterson, C. Zener, Geometric Programming Theory and Applications.  John Wiley and Sons, 1967.

[9] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs.  Mathematical Programming, Vol. 36:307-339, 1986.

[10] F. Erbatur,  O. Hasancebi, I. Tutuncu, H. Kilic (2000) Optimal Design of Planar and Spcae Structures with Genetic Algorithms.  Computers and Structures, 75 (2), 209-224.

[11] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[12] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[13] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[14] Christodoulos A. Floudas, Nonlinear and Mixed-Integer Optimization.  Oxford University Press, 1995.

[15] C. A. Floudas, Deterministic Global Optimization.  Kluwer Academic Publishers, 2000.

[16] S. Gholizadeh, A. Barzegar (2012), Shape Optimization of Structures for Frequency Constraints by Sequential Harmony Search Algorithm, Engineering Optimization, DOI:10.1080/0305215X.2012.704028.

[17] Amos Gilat and Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB.  John Wiley and Sons, Inc. 2008.

[18] Donald Greenspan, Vincenzo Casulli, Numerical Analysis for Applied Mathematics, Science, and Enginerring.  Addison-Wesley Publishing Company, 1988.

[19] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Springer-Verlag 1981.

[20] Majid Jaberipour, Esmaile Khorram (2010), Two Improved Hormony Search Algorithms for Solving Engineering Optimization Problems.  Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3316-3331.

[21] M. Jaberipour, E. Khorram (2010), Solving the Sum-of-Ratios Problems by a Harmony Search Algorithm.  Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[22] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed Discrete Engineering Optimization Problems.  Engineering Optimization, 05/2011, 43:507-523.

[23] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[24] B. K. Kannen , S. N. Kramer (1994), An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design.  Journal of Mechanical Design, 116, pp. 405-411.

[25] R. Baker Kearfott, Some Tests of Generalized Bisection.  ACM Transactions on Mathematical Software, Vol.13, No. 3, September 1987, pages 197-220.

[26] Esmaile Khorram, Majid Jaberipour, Harmony Search Algorithm for Solving Combined Heat and Power Economic Dispatch Problems.  Energy Conversion and Management 52 (2011) 1550-1554.

[27] G. R. Kocis, I. E. Grossmann, Relaxation Strategy for the Structural Optimization of Process Flow Sheets.  Ind. Eng. Chem. Res., 26 (9):1869 (1987).

[28} Sonia Krzyworzcka, Extension of the Lanczos and CGS Methods to Systems of Nonlinear Equations.  Journal of Computational and Aplied Mathematics 69 (1996) 181-190.

[29] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[30] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[31] K. S. Lee, Z. W. Geem (2005), A New Meta-Heuristic Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice.  Computer Methods in Applied Mechanics and Engineering 194 (2005) 3902-3933.

[32] Ya-Zhang Luo, Guo-Jing Tang, Li-Ni Zhou, Hybrid Approach for Solving Systems of Nonlinear Equations Using Chaos Optimization and Quasi-Newton Method.  Applied Soft Computing 8 (2008) 1068-1063.

[33] M. Mahdavi, M. Fesanghary, E. Damangir (2007), An Improved Harmony search Algorithm for Solving Optimization Problems.  Applied Mathematics and Computation, 188 (2), 1567-1579.

[34] C. D. Maranas, C. A. Floudas, Finding All Solutions of Nonlinearly Constrained Systems of Equations.  Journal of Global Optimization, 7(2):143-182, 1995.

[35] Harry M. Markowitz and Alan S. Mann.  On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[36] Keith Meintjes and Alexander P. Morgan, Chemical Equilibrium Systems as Numerical Test Problems.  ACM Transactions on Mathematical Software, Vol. 16, No. 2, June 1990, Pages 143-151.

[37] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[38] Yuanbin Mo, Hetong Liu, Qin Wang,Conjugate Direction Particle Swarm Optimization Solving Systems of Nonlinear Equations. Computers and Mathematics with Applications 57 (2009) 1877-1882.

[39] Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, An Introduction to Interval Analysis.  Cambridge University Press, 2009.

[40] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge University Press, 2000.

[41] W. L. Price, Global Optimization by Controlled Random Search.  Journal of Optimization Theory and Applications, July 1983, Volume 40, Issue 3, pp. 333-348.

[42] Singiresu S. Rao, Ying Xiong (2005), A Hybrid Genetic Algorithm for Mixed-Discrete Design Optimization.  Transaction of the ASME, Vol. 127, November 2005, pp. 1100-1112.

[43] John R. Rice, Numerical Methods, Software, and Analysis, Second Edition.  Academic Press, 1993.

[44] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[45] E. Sandgren (1990), Nonlinear Integer and Discrete Programming in Mechanical Design Optimization.  Journal of Mechanical Design, 112, pp. 223-229.

[46] R. Schinzinger (1965), Optimization in Electromagnetic System Design, in Recent Advances in Optimization Techniques, edited by A. Lavi and T. P. Vogel, John Wiley and Sons, pp. 163-213.

[47] Jung-Fa Tsai (2010), Global Optimization for Signomial Discrete Programming Problems in Engineering Design.  Engineering Optimization, 42:9, pp. 833-843.

[48] L.-W. Tsai, A. P. Morgan, Solving the Kinematics of the Most General Six- and Five-Degree-of Freedom Manipulators by Continuation Methods.  Journal of Mechanisms, Transmissions, and Automation in Design, June 1985, Vol. 107, pp. 189-200.

[49] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[50] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[51] Jsun Yui Wong (2011, May 15).  The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/

[52] Jsun Yui Wong (2011, May 5).  The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/

[53] Jsun Yui Wong (2011, July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[54] Jsun Yui Wong (2012, April 10).  The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design.  Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/

[55] Jsun Yui Wong (2012, April 21).  The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/

[56] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[57] Jsun Yui Wong (2013, January 14).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition.   Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/

[58] Jsun Yui Wong (2013, January 24).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables.   Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/

The Domino Method of General Mixed-Integer Nonlinear Programming Applied to a Very Small Nonlinear Programming Problem from the Literature, Second Edition

Jsun Yui Wong

The following computer program seeks to solve the problem considered by Bracken and McCormick [3], Lee and Geem [31], Jaberipour and Khorram [20, p. 3323, Example 6], among others. One notes Table 5 of Jaberipour and Khorram [20, p. 3323].

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
107 A(1)=-10+FIX(RND*21)
109 A(2)=-10+FIX(RND*21)
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 2
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*1)
140 B=1+FIX(RND*2)
144       IF JJJJ<-31995 THEN  162 ELSE 164
145  REM  IF B=3 OR B=4 GOTO 164
147  REM  IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625                                     ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625                                    ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 2
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
192 X(2)=   (  1    +      X(1)  ) / 2
195 X(3)=     1  -   X(2) ^2    -  X(1)^2   /4
212 IF X(1)<-10 THEN 1670
215 IF X(1)>10 THEN 1670
217 IF X(2)<-10 THEN 1670
218 IF X(2)>10 THEN 1670
221 FOR J44=3 TO 3
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(3)
569  P=      -(X(1)-2     )^2    -(X(2)    -1)^2   +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-4 THEN 1999
1904 PRINT A(1),A(2),A(3)
1989 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The complete output through JJJJ=-31993 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

 .8228756553847113     .9114378276923557   1.952352723932904D-10
-1.393464981049822   -31995  

 .822875655519592      .911437827759796    1.680488387334478D-11
-1.393464980720334   -31994  

 .8228756555315023     .9114378277657512   1.049080960990878D-12
-1.393464980691239   -31993  

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was three seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
 
References

[1] Dana H. Ballard, C. O Jelinek, R. Schinzinger (1974), An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Volume 17, Number 3, pp. 261-266.

[2] M. C. Bartholomew-Biggs, A Numerical Comparison between Two Approaches to the Nonlinear Programming Problem, in Towards Global Optimization 2, edited by L. C. W. Dixon, G. P. Szego, pp. 293-312.  North-Holland Publishing Company 1978.

[3] J. Bracken, G. P. McCormick, Selected Applications of Nonlinear Programming.  New York: John Wiley and Sons, 1968.

[4] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition.  Brooks/Cole 2011.

[5] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.

[6] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[7] K. Deb, M. Goyal (1996), A Combined Genetic Adaptive Search (GeneAS) for Engineering Design.  Computer Science and Informatics, 26 (4), pp. 30-45.

[8] R. J. Duffin, E. L. Peterson, C. Zener, Geometric Programming Theory and Applications.  John Wiley and Sons, 1967.

[9] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs.  Mathematical Programming, Vol. 36:307-339, 1986.

[10] F. Erbatur,  O. Hasancebi, I. Tutuncu, H. Kilic (2000) Optimal Design of Planar and Spcae Structures with Genetic Algorithms.  Computers and Structures, 75 (2), 209-224.

[11] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[12] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[13] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[14] Christodoulos A. Floudas, Nonlinear and Mixed-Integer Optimization.  Oxford University Press, 1995.

[15] C. A. Floudas, Deterministic Global Optimization.  Kluwer Academic Publishers, 2000.

[16] S. Gholizadeh, A. Barzegar (2012), Shape Optimization of Structures for Frequency Constraints by Sequential Harmony Search Algorithm, Engineering Optimization, DOI:10.1080/0305215X.2012.704028.

[17] Amos Gilat and Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB.  John Wiley and Sons, Inc. 2008.

[18] Donald Greenspan, Vincenzo Casulli, Numerical Analysis for Applied Mathematics, Science, and Enginerring.  Addison-Wesley Publishing Company, 1988.

[19] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Springer-Verlag 1981.

[20] Majid Jaberipour, Esmaile Khorram (2010), Two Improved Hormony Search Algorithms for Solving Engineering Optimization Problems.  Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3316-3331.

[21] M. Jaberipour, E. Khorram (2010), Solving the Sum-of-Ratios Problems by a Harmony Search Algorithm.  Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[22] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed Discrete Engineering Optimization Problems.  Engineering Optimization, 05/2011, 43:507-523.

[23] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[24] B. K. Kannen , S. N. Kramer (1994), An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design.  Journal of Mechanical Design, 116, pp. 405-411.

[25] R. Baker Kearfott, Some Tests of Generalized Bisection.  ACM Transactions on Mathematical Software, Vol.13, No. 3, September 1987, pages 197-220.

[26] Esmaile Khorram, Majid Jaberipour, Harmony Search Algorithm for Solving Combined Heat and Power Economic Dispatch Problems.  Energy Conversion and Management 52 (2011) 1550-1554.

[27] G. R. Kocis, I. E. Grossmann, Relaxation Strategy for the Structural Optimization of Process Flow Sheets.  Ind. Eng. Chem. Res., 26 (9):1869 (1987).

[28} Sonia Krzyworzcka, Extension of the Lanczos and CGS Methods to Systems of Nonlinear Equations.  Journal of Computational and Aplied Mathematics 69 (1996) 181-190.

[29] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[30] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[31] K. S. Lee, Z. W. Geem (2005), A New Meta-Heuristic Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice.  Computer Methods in Applied Mechanics and Engineering 194 (2005) 3902-3933.

[32] Ya-Zhang Luo, Guo-Jing Tang, Li-Ni Zhou, Hybrid Approach for Solving Systems of Nonlinear Equations Using Chaos Optimization and Quasi-Newton Method.  Applied Soft Computing 8 (2008) 1068-1063.

[33] M. Mahdavi, M. Fesanghary, E. Damangir (2007), An Improved Harmony search Algorithm for Solving Optimization Problems.  Applied Mathematics and Computation, 188 (2), 1567-1579.

[34] C. D. Maranas, C. A. Floudas, Finding All Solutions of Nonlinearly Constrained Systems of Equations.  Journal of Global Optimization, 7(2):143-182, 1995.

[35] Harry M. Markowitz and Alan S. Mann.  On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[36] Keith Meintjes and Alexander P. Morgan, Chemical Equilibrium Systems as Numerical Test Problems.  ACM Transactions on Mathematical Software, Vol. 16, No. 2, June 1990, Pages 143-151.

[37] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[38] Yuanbin Mo, Hetong Liu, Qin Wang,Conjugate Direction Particle Swarm Optimization Solving Systems of Nonlinear Equations. Computers and Mathematics with Applications 57 (2009) 1877-1882.

[39] Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, An Introduction to Interval Analysis.  Cambridge University Press, 2009.

[40] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge University Press, 2000.

[41] W. L. Price, Global Optimization by Controlled Random Search.  Journal of Optimization Theory and Applications, July 1983, Volume 40, Issue 3, pp. 333-348.

[42] Singiresu S. Rao, Ying Xiong (2005), A Hybrid Genetic Algorithm for Mixed-Discrete Design Optimization.  Transaction of the ASME, Vol. 127, November 2005, pp. 1100-1112.

[43] John R. Rice, Numerical Methods, Software, and Analysis, Second Edition.  Academic Press, 1993.

[44] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[45] E. Sandgren (1990), Nonlinear Integer and Discrete Programming in Mechanical Design Optimization.  Journal of Mechanical Design, 112, pp. 223-229.

[46] R. Schinzinger (1965), Optimization in Electromagnetic System Design, in Recent Advances in Optimization Techniques, edited by A. Lavi and T. P. Vogel, John Wiley and Sons, pp. 163-213.

[47] Jung-Fa Tsai (2010), Global Optimization for Signomial Discrete Programming Problems in Engineering Design.  Engineering Optimization, 42:9, pp. 833-843.

[48] L.-W. Tsai, A. P. Morgan, Solving the Kinematics of the Most General Six- and Five-Degree-of Freedom Manipulators by Continuation Methods.  Journal of Mechanisms, Transmissions, and Automation in Design, June 1985, Vol. 107, pp. 189-200.

[49] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[50] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[51] Jsun Yui Wong (2011, May 15).  The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/

[52] Jsun Yui Wong (2011, May 5).  The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/

[53] Jsun Yui Wong (2011, July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[54] Jsun Yui Wong (2012, April 10).  The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design.  Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/

[55] Jsun Yui Wong (2012, April 21).  The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/

[56] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[57] Jsun Yui Wong (2013, January 14).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition.   Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/

[58] Jsun Yui Wong (2013, January 24).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables.   Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/

The Domino Method of General Mixed-Integer Nonlinear Programming Applied to a Very Small Nonlinear Programming Problem from the Literature

Jsun Yui Wong

The following computer program seeks to solve the problem considered by Bracken and McCormick [44], Lee and Geem [30], Jaberipour and Khorram [19, p. 3323, Example 6], among others.

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
107 A(1)=-10+FIX(RND*21)
109 A(2)=-10+FIX(RND*21)
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 2
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*1)
140 B=1+FIX(RND*2)
144       IF JJJJ<-31995 THEN  162 ELSE 164
145  REM  IF B=3 OR B=4 GOTO 164
147  REM  IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625                                     ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625                                    ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 2
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
192 X(2)=   (  1    +      X(1)  ) / 2
195 X(3)=     1  -   X(2) ^2    -  X(1)^2   /4
212 IF X(1)<-10 THEN 1670
215 IF X(1)>10 THEN 1670
217 IF X(2)<-10 THEN 1670
218 IF X(2)>10 THEN 1670
221 FOR J44=3 TO 3
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(3)
569  P=      -(X(1)-2     )^2    -(X(2)    -1)^2   +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 3
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-4 THEN 1999
1904 PRINT A(1),A(2),A(3)
1989 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The complete output through JJJJ=-31993 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

 .8228756553847113     .9114378276923557   1.952352723932904D-10
-1.393464981049822   -31995  

 .822875655519592      .911437827759796    1.680488387334478D-11
-1.393464980720334   -31994  

 .8228756555315023     .9114378277657512   1.049080960990878D-12
-1.393464980691239   -31993  

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was three seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
 
References

[1] Dana H. Ballard, C. O Jelinek, R. Schinzinger (1974), An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Volume 17, Number 3, pp. 261-266.

[2] M. C. Bartholomew-Biggs, A Numerical Comparison between Two Approaches to the Nonlinear Programming Problem, in Towards Global Optimization 2, edited by L. C. W. Dixon, G. P. Szego, pp. 293-312.  North-Holland Publishing Company 1978.

[3] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition.  Brooks/Cole 2011.

[4] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.

[5] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[6] K. Deb, M. Goyal (1996), A Combined Genetic Adaptive Search (GeneAS) for Engineering Design.  Computer Science and Informatics, 26 (4), pp. 30-45.

[7] R. J. Duffin, E. L. Peterson, C. Zener, Geometric Programming Theory and Applications.  John Wiley and Sons, 1967.

[8] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs.  Mathematical Programming, Vol. 36:307-339, 1986.

[9] F. Erbatur,  O. Hasancebi, I. Tutuncu, H. Kilic (2000) Optimal Design of Planar and Spcae Structures with Genetic Algorithms.  Computers and Structures, 75 (2), 209-224.

[10] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[11] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[12] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[32] Christodoulos A. Floudas, Nonlinear and Mixed-Integer Optimization.  Oxford University Press, 1995.

[14] C. A. Floudas, Deterministic Global Optimization.  Kluwer Academic Publishers, 2000.

[15] S. Gholizadeh, A. Barzegar (2012), Shape Optimization of Structures for Frequency Constraints by Sequential Harmony Search Algorithm, Engineering Optimization, DOI:10.1080/0305215X.2012.704028.

[16] Amos Gilat and Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB.  John Wiley and Sons, Inc. 2008.

[17] Donald Greenspan, Vincenzo Casulli, Numerical Analysis for Applied Mathematics, Science, and Enginerring.  Addison-Wesley Publishing Company, 1988.

[18] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Springer-Verlag 1981.

[19] Majid Jaberipour, Esmaile Khorram (2010), Two Improved Hormony Search Algorithms for Solving Engineering Optimization Problems.  Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3316-3331.

[20] M. Jaberipour, E. Khorram (2010), Solving the Sum-of-Ratios Problems by a Harmony Search Algorithm.  Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[21] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed Discrete Engineering Optimization Problems.  Engineering Optimization, 05/2011, 43:507-523.

[22] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[23] B. K. Kannen , S. N. Kramer (1994), An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design.  Journal of Mechanical Design, 116, pp. 405-411.

[24] R. Baker Kearfott, Some Tests of Generalized Bisection.  ACM Transactions on Mathematical Software, Vol.13, No. 3, September 1987, pages 197-220.

[25] Esmaile Khorram, Majid Jaberipour, Harmony Search Algorithm for Solving Combined Heat and Power Economic Dispatch Problems.  Energy Conversion and Management 52 (2011) 1550-1554.

[26] G. R. Kocis, I. E. Grossmann, Relaxation Strategy for the Structural Optimization of Process Flow Sheets.  Ind. Eng. Chem. Res., 26 (9):1869 (1987).

[27} Sonia Krzyworzcka, Extension of the Lanczos and CGS Methods to Systems of Nonlinear Equations.  Journal of Computational and Aplied Mathematics 69 (1996) 181-190.

[28] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[29] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[30] K. S. Lee, Z. W. Geem (2005), A New Meta-Heuristic Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice.  Computer Methods in Applied Mechanics and Engineering 194 (2005) 3902-3933.

[31] Ya-Zhang Luo, Guo-Jing Tang, Li-Ni Zhou, Hybrid Approach for Solving Systems of Nonlinear Equations Using Chaos Optimization and Quasi-Newton Method.  Applied Soft Computing 8 (2008) 1068-1063.

[32] M. Mahdavi, M. Fesanghary, E. Damangir (2007), An Improved Harmony search Algorithm for Solving Optimization Problems.  Applied Mathematics and Computation, 188 (2), 1567-1579.

[33] C. D. Maranas, C. A. Floudas, Finding All Solutions of Nonlinearly Constrained Systems of Equations.  Journal of Global Optimization, 7(2):143-182, 1995.

[34] Harry M. Markowitz and Alan S. Mann.  On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[35] Keith Meintjes and Alexander P. Morgan, Chemical Equilibrium Systems as Numerical Test Problems.  ACM Transactions on Mathematical Software, Vol. 16, No. 2, June 1990, Pages 143-151.

[36] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[37] Yuanbin Mo, Hetong Liu, Qin Wang,Conjugate Direction Particle Swarm Optimization Solving Systems of Nonlinear Equations. Computers and Mathematics with Applications 57 (2009) 1877-1882.

[38] Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, An Introduction to Interval Analysis.  Cambridge University Press, 2009.

[39] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge University Press, 2000.

[40] W. L. Price, Global Optimization by Controlled Random Search.  Journal of Optimization Theory and Applications, July 1983, Volume 40, Issue 3, pp. 333-348.

[41] Singiresu S. Rao, Ying Xiong (2005), A Hybrid Genetic Algorithm for Mixed-Discrete Design Optimization.  Transaction of the ASME, Vol. 127, November 2005, pp. 1100-1112.

[42] John R. Rice, Numerical Methods, Software, and Analysis, Second Edition.  Academic Press, 1993.

[43] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[44] E. Sandgren (1990), Nonlinear Integer and Discrete Programming in Mechanical Design Optimization.  Journal of Mechanical Design, 112, pp. 223-229.

[45] R. Schinzinger (1965), Optimization in Electromagnetic System Design, in Recent Advances in Optimization Techniques, edited by A. Lavi and T. P. Vogel, John Wiley and Sons, pp. 163-213.

[46] Jung-Fa Tsai (2010), Global Optimization for Signomial Discrete Programming Problems in Engineering Design.  Engineering Optimization, 42:9, pp. 833-843.

[47] L.-W. Tsai, A. P. Morgan, Solving the Kinematics of the Most General Six- and Five-Degree-of Freedom Manipulators by Continuation Methods.  Journal of Mechanisms, Transmissions, and Automation in Design, June 1985, Vol. 107, pp. 189-200.

[48] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[49] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[50] Jsun Yui Wong (2011, May 15).  The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/

[51] Jsun Yui Wong (2011, May 5).  The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/

[52] Jsun Yui Wong (2011, July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[53] Jsun Yui Wong (2012, April 10).  The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design.  Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/

[54] Jsun Yui Wong (2012, April 21).  The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/

[55] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[56] Jsun Yui Wong (2013, January 14).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition.   Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/

[57] Jsun Yui Wong (2013, January 24).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables.   Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/

The Domino Method of General Mixed-Integer Nonlinear Programming Applied to Pressure Vessel Design Involving Discrete Variables and Continuous Variables, Second Edition

Jsun Yui Wong

The following computer program seeks to solve the problem considered by Sandgren [44], Lee and Geem [30], Mahdavi, Fesanghary, and Damangir [32], Jaberipour and Khorram [19], and Gholizadeh and Barzegar [15], among others. This problem has discrete variables and continuous variables.

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144       IF JJJJ<-31995 THEN  162 ELSE 145
145       IF B=3 OR B=4 GOTO 164
147       IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625                                     ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625                                    ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 4
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
194    X(5)=X(1)        -.0193*X(3)
195 X(6)=X(2)         -.00954*X(3)
199 X(7)=-1296000!     + 1.3333333333333#*3.141592654#*X(3)^3                    +        3.141592654#*X(3) ^2   *X(4)
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569  P=      -.6224*X(1)*X(3)*X(4)    -1.7781*X(2)*X(3) ^2   -3.1611*X(1)   ^2   *X(4)      -19.84*X(1)  ^2   *X(3)    +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The top-two candidate solutions through JJJJ=-30062 are shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

1.125   .625   58.28628654289647
43.71393346695399   7.462646913716786D-05   6.894882643287964D-02
1.736438644002192D-04     -7197.965676984811   -31656

1.125   .625               58.28628935954246
43.71391795959724          7.457210786745994D-05   6.894879956207689D-02
4.156150680501014D-06     -7197.965503888046   -30062

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was ten minutes.

The following computer program tries to improve on the solution shown above at JJJJ=-30062.  The A(7)= 4.156150680501014D-06 shown above suggests that at optimality it is likely that A(7)=0; in other words, the corresponding constraint is active.  That is tried with line 200 of the following computer program.    
            
While line 199 of the computer program above is
199 X(7)=-1296000!     + 1.3333333333333#*3.141592654#*X(3)^3                    +        3.141592654#*X(3) ^2   *X(4),
line 199 below is
199 REM X(7)=-1296000!     + 1.3333333333333#*3.141592654#*X(3)^3                    +        3.141592654#*X(3) ^2   *X(4).

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144       IF JJJJ<-31995 THEN  162 ELSE 145
145       IF B=3 OR B=4 GOTO 164
147       IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625                                     ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625                                    ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 4
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
194    X(5)=X(1)        -.0193*X(3)
195 X(6)=X(2)         -.00954*X(3)
199 REM X(7)=-1296000!     + 1.3333333333333#*3.141592654#*X(3)^3                    +        3.141592654#*X(3) ^2   *X(4)
200 X(4)=    (    +    1296000!     - 1.3333333333333#*3.141592654#*X(3)^3  )             /  (+        3.141592654#*X(3) ^2  )
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569  P=      -.6224*X(1)*X(3)*X(4)    -1.7781*X(2)*X(3) ^2   -3.1611*X(1)   ^2   *X(4)      -19.84*X(1)  ^2   *X(3)    +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The complete output through JJJJ=-31988 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

1.125   .625               58.29015090845521
43.69268114487037          4.421098581675054D-08   6.891196038545272D-02
0                         -7197.729199130497   -31995

1.125   .625               58.29015308692593
43.69266916514552          2.166499452060933D-09   6.891193960284212D-02
0                         -7197.729065837913   -31994

1.125   .625               58.29013414146686
43.69277334899321          3.678138735530023D-07   6.891212034252163D-02
0                         -7197.730225041295   -31991

1.125   .625               58.2901531892509
43.69266860244574          1.916273528745904D-10   6.891193862666185D-02
0                         -7197.729059577026   -31988

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was four seconds.

Similarly the residual A(5)= 7.462646913716786D-05 at JJJJ=31656 of the first computer program listed above suggests that at optimality it is likely that A(5)=0; in other words, the corresponding constraint is active.  That is tried with line 192 of the following computer program; the following line 194 is just 194 REM   X(5)=X(1) -.0193*X(3).

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144       IF JJJJ<-31995 THEN  162 ELSE 145
145       IF B=3 OR B=4 GOTO 164
147       IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625                                     ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625                                    ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 4
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
192 X(3)=X(1)   /    .0193
194 REM   X(5)=X(1)        -.0193*X(3)
195 X(6)=X(2)         -.00954*X(3)
199 X(7)=-1296000!     + 1.3333333333333#*3.141592654#*X(3)^3                    +        3.141592654#*X(3) ^2   *X(4)
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569  P=      -.6224*X(1)*X(3)*X(4)    -1.7781*X(2)*X(3) ^2   -3.1611*X(1)   ^2   *X(4)      -19.84*X(1)  ^2   *X(3)    +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The complete output through JJJJ=-31979 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

1.125   .625                   58.29015319917978
43.69814366322346          0   6.891193853194036D-02
58.44314005206252             -7197.974429174728   -31992

1.125   .625                   58.29015319917978
43.69266979995663          0   6.891193853194036D-02
1.336543742218055D-02         -7197.729115083545   -31984

1.125   .625                   58.29015319917978
43.69266942413557          0   6.891193853194036D-02
9,353802692203317D-03         -7197.729098240924   -31983

1.125   .625                   58.29015319917978
43.69266883596002          0   6.891193853194036D-02
3.075427965086419D-03         -7197.729071881524   -31979
  
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was five seconds.

In conclusion, guessing correctly before or after a computer run which slack variable equals zero at optimality can shorten computational time.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
 
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[48] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[49] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[50] Jsun Yui Wong (2011, May 15).  The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/

[51] Jsun Yui Wong (2011, May 5).  The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/

[52] Jsun Yui Wong (2011, July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[53] Jsun Yui Wong (2012, April 10).  The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design.  Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/

[54] Jsun Yui Wong (2012, April 21).  The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/

[55] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[56] Jsun Yui Wong (2013, January 14).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition.   Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/

[57] Jsun Yui Wong (2013, January 24).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables.   Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/

The Domino Method of General Mixed-Integer Nonlinear Programming Applied to Pressure Vessel Design Involving Discrete Variables and Continuous Variables

Jsun Yui Wong

The following computer program seeks to solve the problem considered by Sandgren [44], Lee and Geem [30], Mahdavi, Fesanghary, and Damangir [32], Jaberipour and Khorram [19], and Gholizadeh and Barzegar [15], among others. This problem has discrete variables and continuous variables.

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144       IF JJJJ<-31995 THEN  162 ELSE 145
145       IF B=3 OR B=4 GOTO 164
147       IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625                                     ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625                                    ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 4
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
194    X(5)=X(1)        -.0193*X(3)
195 X(6)=X(2)         -.00954*X(3)
199 X(7)=-1296000!     + 1.3333333333333#*3.141592654#*X(3)^3                    +        3.141592654#*X(3) ^2   *X(4)
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569  P=      -.6224*X(1)*X(3)*X(4)    -1.7781*X(2)*X(3) ^2   -3.1611*X(1)   ^2   *X(4)      -19.84*X(1)  ^2   *X(3)    +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The top-two candidate solutions through JJJJ=-30062 are shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

1.125   .625   58.28628654289647
43.71393346695399   7.462646913716786D-05   6.894882643287964D-02
1.736438644002192D-04     -7197.965676984811   -31656

1.125   .625               58.28628935954246
43.71391795959724          7.457210786745994D-05   6.894879956207689D-02
4.156150680501014D-06     -7197.965503888046   -30062

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was ten minutes.

The following computer program tries to improve on the solution shown above at JJJJ=-30062.  The A(7)= 4.156150680501014D-06 shown above suggests that at optimality it is likely that A(7)=0; in other words, the corresponding constraint is active.  That is tried with line 200 of the following computer program.    
            
While line 199 of the computer program above is
199 X(7)=-1296000!     + 1.3333333333333#*3.141592654#*X(3)^3                    +        3.141592654#*X(3) ^2   *X(4),
line 199 below is
199 REM X(7)=-1296000!     + 1.3333333333333#*3.141592654#*X(3)^3                    +        3.141592654#*X(3) ^2   *X(4).

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144       IF JJJJ<-31995 THEN  162 ELSE 145
145       IF B=3 OR B=4 GOTO 164
147       IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625                                     ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625                                    ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162   IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE    X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 4
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
194    X(5)=X(1)        -.0193*X(3)
195 X(6)=X(2)         -.00954*X(3)
199 REM X(7)=-1296000!     + 1.3333333333333#*3.141592654#*X(3)^3                    +        3.141592654#*X(3) ^2   *X(4)
200 X(4)=    (    +    1296000!     - 1.3333333333333#*3.141592654#*X(3)^3  )             /  (+        3.141592654#*X(3) ^2  )
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569  P=      -.6224*X(1)*X(3)*X(4)    -1.7781*X(2)*X(3) ^2   -3.1611*X(1)   ^2   *X(4)      -19.84*X(1)  ^2   *X(3)    +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The complete output through JJJJ=-31988 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

1.125   .625               58.29015090845521
43.69268114487037          4.421098581675054D-08   6.891196038545272D-02
0                         -7197.729199130497   -31995

1.125   .625               58.29015308692593
43.69266916514552          2.166499452060933D-09   6.891193960284212D-02
0                         -7197.729065837913   -31994

1.125   .625               58.29013414146686
43.69277334899321          3.678138735530023D-07   6.891212034252163D-02
0                         -7197.730225041295   -31991

1.125   .625               58.2901531892509
43.69266860244574          1.916273528745904D-10   6.891193862666185D-02
0                         -7197.729059577026   -31988

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was 4 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
 
References

[1] Dana H. Ballard, C. O Jelinek, R. Schinzinger (1974), An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Volume 17, Number 3, pp. 261-266.

[2] M. C. Bartholomew-Biggs, A Numerical Comparison between Two Approaches to the Nonlinear Programming Problem, in Towards Global Optimization 2, edited by L. C. W. Dixon, G. P. Szego, pp. 293-312.  North-Holland Publishing Company 1978.

[3] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition.  Brooks/Cole 2011.

[4] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.

[5] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[6] K. Deb, M. Goyal (1996), A Combined Genetic Adaptive Search (GeneAS) for Engineering Design.  Computer Science and Informatics, 26 (4), pp. 30-45.

[7] R. J. Duffin, E. L. Peterson, C. Zener, Geometric Programming Theory and Applications.  John Wiley and Sons, 1967.

[8] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs.  Mathematical Programming, Vol. 36:307-339, 1986.

[9] F. Erbatur,  O. Hasancebi, I. Tutuncu, H. Kilic (2000) Optimal Design of Planar and Spcae Structures with Genetic Algorithms.  Computers and Structures, 75 (2), 209-224.

[10] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[11] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[12] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[32] Christodoulos A. Floudas, Nonlinear and Mixed-Integer Optimization.  Oxford University Press, 1995.

[14] C. A. Floudas, Deterministic Global Optimization.  Kluwer Academic Publishers, 2000.

[15] S. Gholizadeh, A. Barzegar (2012), Shape Optimization of Structures for Frequency Constraints by Sequential Harmony Search Algorithm, Engineering Optimization, DOI:10.1080/0305215X.2012.704028.

[16] Amos Gilat and Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB.  John Wiley and Sons, Inc. 2008.

[17] Donald Greenspan, Vincenzo Casulli, Numerical Analysis for Applied Mathematics, Science, and Enginerring.  Addison-Wesley Publishing Company, 1988.

[18] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Springer-Verlag 1981.

[19] Majid Jaberipour, Esmaile Khorram (2010), Two Improved Hormony Search Algorithms for Solving Engineering Optimization Problems.  Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3316-3331.

[20] M. Jaberipour, E. Khorram (2010), Solving the Sum-of-Ratios Problems by a Harmony Search Algorithm.  Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[21] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed Discrete Engineering Optimization Problems.  Engineering Optimization, 05/2011, 43:507-523.

[22] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[23] B. K. Kannen , S. N. Kramer (1994), An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design.  Journal of Mechanical Design, 116, pp. 405-411.

[24] R. Baker Kearfott, Some Tests of Generalized Bisection.  ACM Transactions on Mathematical Software, Vol.13, No. 3, September 1987, pages 197-220.

[25] Esmaile Khorram, Majid Jaberipour, Harmony Search Algorithm for Solving Combined Heat and Power Economic Dispatch Problems.  Energy Conversion and Management 52 (2011) 1550-1554.

[26] G. R. Kocis, I. E. Grossmann, Relaxation Strategy for the Structural Optimization of Process Flow Sheets.  Ind. Eng. Chem. Res., 26 (9):1869 (1987).

[27} Sonia Krzyworzcka, Extension of the Lanczos and CGS Methods to Systems of Nonlinear Equations.  Journal of Computational and Aplied Mathematics 69 (1996) 181-190.

[28] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[29] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[30] K. S. Lee, Z. W. Geem (2005), A New Meta-Heuristic Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice.  Computer Methods in Applied Mechanics and Engineering 194 (2005) 3902-3933.

[31] Ya-Zhang Luo, Guo-Jing Tang, Li-Ni Zhou, Hybrid Approach for Solving Systems of Nonlinear Equations Using Chaos Optimization and Quasi-Newton Method.  Applied Soft Computing 8 (2008) 1068-1063.

[32] M. Mahdavi, M. Fesanghary, E. Damangir (2007), An Improved Harmony search Algorithm for Solving Optimization Problems.  Applied Mathematics and Computation, 188 (2), 1567-1579.

[33] C. D. Maranas, C. A. Floudas, Finding All Solutions of Nonlinearly Constrained Systems of Equations.  Journal of Global Optimization, 7(2):143-182, 1995.

[34] Harry M. Markowitz and Alan S. Mann.  On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[35] Keith Meintjes and Alexander P. Morgan, Chemical Equilibrium Systems as Numerical Test Problems.  ACM Transactions on Mathematical Software, Vol. 16, No. 2, June 1990, Pages 143-151.

[36] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[37] Yuanbin Mo, Hetong Liu, Qin Wang,Conjugate Direction Particle Swarm Optimization Solving Systems of Nonlinear Equations. Computers and Mathematics with Applications 57 (2009) 1877-1882.

[38] Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, An Introduction to Interval Analysis.  Cambridge University Press, 2009.

[39] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge University Press, 2000.

[40] W. L. Price, Global Optimization by Controlled Random Search.  Journal of Optimization Theory and Applications, July 1983, Volume 40, Issue 3, pp. 333-348.

[41] Singiresu S. Rao, Ying Xiong (2005), A Hybrid Genetic Algorithm for Mixed-Discrete Design Optimization.  Transaction of the ASME, Vol. 127, November 2005, pp. 1100-1112.

[42] John R. Rice, Numerical Methods, Software, and Analysis, Second Edition.  Academic Press, 1993.

[43] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[44] E. Sandgren (1990), Nonlinear Integer and Discrete Programming in Mechanical Design Optimization.  Journal of Mechanical Design, 112, pp. 223-229.

[45] R. Schinzinger (1965), Optimization in Electromagnetic System Design, in Recent Advances in Optimization Techniques, edited by A. Lavi and T. P. Vogel, John Wiley and Sons, pp. 163-213.

[46] Jung-Fa Tsai (2010), Global Optimization for Signomial Discrete Programming Problems in Engineering Design.  Engineering Optimization, 42:9, pp. 833-843.

[47] L.-W. Tsai, A. P. Morgan, Solving the Kinematics of the Most General Six- and Five-Degree-of Freedom Manipulators by Continuation Methods.  Journal of Mechanisms, Transmissions, and Automation in Design, June 1985, Vol. 107, pp. 189-200.

[48] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[49] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[50] Jsun Yui Wong (2011, May 15).  The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/

[51] Jsun Yui Wong (2011, May 5).  The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/

[52] Jsun Yui Wong (2011, July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[53] Jsun Yui Wong (2012, April 10).  The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design.  Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/

[54] Jsun Yui Wong (2012, April 21).  The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/

[55] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[56] Jsun Yui Wong (2013, January 14).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition.   Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/

[57] Jsun Yui Wong (2013, January 24).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables.   Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/

The Domino Method of General Mixed-Integer Nonlinear Programming Applied to Transformer Design, Fourth Edition

Jsun Yui Wong

The computer program of the paper posted on April 18 2013 seeks to solve the problem considered by Ballard, Jelinek, and Schinzinger [1, p. 265, Problem 3], Bartholomew-Biggs [2, p. 307], Hock and Schittkowski [17, page 108, Problem 93], and Price [37].  For convenience, the best solution of that April 18 paper is shown below:

 5.21594626470976       4.679281609886584    10.4995363811351
 12.24139625929538      .7575198783146843    .8710897559328291
 8.968492615224477D-12  3.827390190292157D-04
-135.1206569520764     -30977

The computer program listed below tries to improve on the solution shown above. The A(7)= 8.968492615224477D-12 shown above suggests that at optimality it is likely that A(7)=0; in other words, the corresponding constraint is active.  That is tried with line 192 of the following computer program.  This line 192 is new and is 192 X(6)= 2.07/(+.001*X(1)*X(2)*X(3)*X(4)*X(5)    ).
               
While line 194 of the earlier paper is 194 X(7)=-2.07+.001*X(1)*X(2)*X(3)*X(4)*X(5)*X(6),
here line 194 is 194 REM  X(7)=-2.07+.001*X(1)*X(2)*X(3)*X(4)*X(5)*X(6).

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
100 FOR KLQ=1 TO 6
101 A(KLQ)=1+FIX(RND*10)
102 NEXT KLQ
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 6
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
159 FOR IPP=1 TO FIX(1+RND*3)
160 B=1+FIX(RND*6)
161 IF JJJJ<-31995 THEN  162 ELSE 164
162 IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE       X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 6
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
191 IF        (+.001*X(1)*X(2)*X(3)*X(4)*X(5)       )  <1E-08 THEN 1670
192 X(6)=        2.07/(+.001*X(1)*X(2)*X(3)*X(4)*X(5)       )
194 REM   X(7)=-2.07+.001*X(1)*X(2)*X(3)*X(4)*X(5)*X(6)
195 X(8)=-.00062*X(1)*X(4)*X(5)^2 *(  X(1)+X(2)+X(3)     )    -.00058*X(2)*X(3)      *X(6)^2    *(X(1)+1.57*X(2)+X(4)    )       +1
211 FOR J44=1 TO 6
212 IF X(J44)<0 THEN 1670
214 NEXT J44
221 FOR J44=7 TO 8
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(7)        +P(8)
569  P=-.0204*X(1)*X(4)*(X(1)+X(2)+X(3)     )    -.0187*X(2)*X(3)*(X(1)+1.57*X(2)+X(4)    )      -.0607*X(1)*X(4)*X(5)^2 *(X(1)+X(2)+X(3)  ) -.0437*X(2)*X(3)*X(6)^2  *(X(1)+1.57*X(2)+X(4)      )   +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 8
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-135.09 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1908 PRINT A(7),A(8)
1989 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The complete output through JJJJ=-31807 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

 5.350621215941249   4.730358613525082   10.3692826026934
 11.95985253304095   .752283489916058    .8766292774891455
 0   5.301453720463201D-13
-135.0899657638963   -31833

 5.386409040659636   4.637267386602819   10.41685471794954
 12.03392137292143   .7489440948583807   .8827051585109831
 0   2.910172103298692D-13
-135.08080596285   -31814

 5.336609240984221   4.633232482838074   10.44174204694643
 12.1052622688247    .7536013782356158   .8788848272984081
 0   5.934142066621462D-14
-135.0784572650097   -31807

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was eight minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
 
References

[1] Dana H. Ballard, C. O Jelinek, R. Schinzinger (1974), An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Volume 17, Number 3, pp. 261-266.

[2] M. C. Bartholomew-Biggs, A Numerical Comparison between Two Approaches to the Nonlinear Programming Problem, in Towards Global Optimization 2, edited by L. C. W. Dixon, G. P. Szego, pp. 293-312.  North-Holland Publishing Company 1978.

[3] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition.  Brooks/Cole 2011.

[4] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.

[5] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[6] K. Deb, M. Goyal (1996), A Combined Genetic Adaptive Search (GeneAS) for Engineering Design.  Computer Science and Informatics, 26 (4), pp. 30-45.

[7] R. J. Duffin, E. L. Peterson, C. Zener, Geometric Programming Theory and Applications.  John Wiley and Sons, 1967.

[8] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs.  Mathematical Programming, Vol. 36:307-339, 1986.

[9] F. Erbatur,  O. Hasancebi, I. Tutuncu, H. Kilic (2000) Optimal Design of Planar and Spcae Structures with Genetic Algorithms.  Computers and Structures, 75 (2), 209-224.

[10] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[11] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[12] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[32] Christodoulos A. Floudas, Nonlinear and Mixed-Integer Optimization.  Oxford University Press, 1995.

[14] C. A. Floudas, Deterministic Global Optimization.  Kluwer Academic Publishers, 2000.

[15] Amos Gilat and Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB.  John Wiley and Sons, Inc. 2008.

[16] Donald Greenspan, Vincenzo Casulli, Numerical Analysis for Applied Mathematics, Science, and Enginerring.  Addison-Wesley Publishing Company, 1988.

[17] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Springer-Verlag 1981.

[18] Majid Jaberipour, Esmaile Khorram (2010), Two Improved Hormony Search Algorithms for Solving Engineering Optimization Problems.  Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3316-3331.

[19] M. Jaberipour, E. Khorram (2010), Solving the Sum-of-Ratios Problems by a Harmony Search Algorithm.  Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[20] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed Discrete Engineering Optimization Problems.  Engineering Optimization, 05/2011, 43:507-523.

[21] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[22] B. K. Kannen , S. N. Kramer (1994), An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design.  Journal of Mechanical Design, 116, pp. 405-411.

[23] R. Baker Kearfott, Some Tests of Generalized Bisection.  ACM Transactions on Mathematical Software, Vol.13, No. 3, September 1987, pages 197-220.

[24] Esmaile Khorram, Majid Jaberipour, Harmony Search Algorithm for Solving Combined Heat and Power Economic Dispatch Problems.  Energy Conversion and Management 52 (2011) 1550-1554.

[25] G. R. Kocis, I. E. Grossmann, Relaxation Strategy for the Structural Optimization of Process Flow Sheets.  Ind. Eng. Chem. Res., 26 (9):1869 (1987).

[26} Sonia Krzyworzcka, Extension of the Lanczos and CGS Methods to Systems of Nonlinear Equations.  Journal of Computational and Aplied Mathematics 69 (1996) 181-190.

[27] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[28] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[29] Ya-Zhang Luo, Guo-Jing Tang, Li-Ni Zhou, Hybrid Approach for Solving Systems of Nonlinear Equations Using Chaos Optimization and Quasi-Newton Method.  Applied Soft Computing 8 (2008) 1068-1063.

[30] C. D. Maranas, C. A. Floudas, Finding All Solutions of Nonlinearly Constrained Systems of Equations.  Journal of Global Optimization, 7(2):143-182, 1995.

[31] Harry M. Markowitz and Alan S. Mann.  On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[32] Keith Meintjes and Alexander P. Morgan, Chemical Equilibrium Systems as Numerical Test Problems.  ACM Transactions on Mathematical Software, Vol. 16, No. 2, June 1990, Pages 143-151.

[33] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[34] Yuanbin Mo, Hetong Liu, Qin Wang,Conjugate Direction Particle Swarm Optimization Solving Systems of Nonlinear Equations. Computers and Mathematics with Applications 57 (2009) 1877-1882.

[35] Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, An Introduction to Interval Analysis.  Cambridge University Press, 2009.

[36] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge University Press, 2000.

[37] W. L. Price, Global Optimization by Controlled Random Search.  Journal of Optimization Theory and Applications, July 1983, Volume 40, Issue 3, pp. 333-348.

[38] Singiresu S. Rao, Ying Xiong (2005), A Hybrid Genetic Algorithm for Mixed-Discrete Design Optimization.  Transaction of the ASME, Vol. 127, November 2005, pp. 1100-1112.

[39] John R. Rice, Numerical Methods, Software, and Analysis, Second Edition.  Academic Press, 1993.

[40] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[41] E. Sandgren (1990), Nonlinear Integer and Discrete Programming in Mechanical Design Optimization.  Journal of Mechanical Design, 112, pp. 223-229.

[42] R. Schinzinger (1965), Optimization in Electromagnetic System Design, in Recent Advances in Optimization Techniques, edited by A. Lavi and T. P. Vogel, John Wiley and Sons, pp. 163-213.

[43] Jung-Fa Tsai (2010), Global Optimization for Signomial Discrete Programming Problems in Engineering Design.  Engineering Optimization, 42:9, pp. 833-843.

[44] L.-W. Tsai, A. P. Morgan, Solving the Kinematics of the Most General Six- and Five-Degree-of Freedom Manipulators by Continuation Methods.  Journal of Mechanisms, Transmissions, and Automation in Design, June 1985, Vol. 107, pp. 189-200.

[45] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[46] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[47] Jsun Yui Wong (2011, May 15).  The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/

[48] Jsun Yui Wong (2011, May 5).  The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/

[49] Jsun Yui Wong (2011, July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[50] Jsun Yui Wong (2012, April 10).  The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design.  Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/

[51] Jsun Yui Wong (2012, April 21).  The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/

[52] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[53] Jsun Yui Wong (2013, January 14).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition.   Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/

[54] Jsun Yui Wong (2013, January 24).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables.   Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/

The Domino Method of General Mixed-Integer Nonlinear Programming Applied to Transformer Design, Third Edition

Jsun Yui Wong

The following computer program tries to improve on the solutions of the preceding paper.

For convenience, the best solution of the preceding paper is shown below:

 5   4   9
 10   .9454661009311858   1.216331190689866
 2.147976241317906D-11    4.383658270016441D-02
-131.0487522710977   -31806

The A(7)= 2.147976241317906D-11 shown above suggests that at optimality A(7)=0 is likely.  That is tried with line 184, which is 184 X(6)=(2.07            )/  (    +.001*X(1)*X(2)*X(3)*X(4)*X(5)   ), of the following computer program.

While line 194 of the preceding paper is 194 X(7)=-2.07+.001*X(1)*X(2)*X(3)*X(4)*X(5)*X(6),
here line 194 is 194 REM  X(7)=-2.07+.001*X(1)*X(2)*X(3)*X(4)*X(5)*X(6).

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM    DIM PE(35,35)
5 REM    DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
100 FOR KLQ=1 TO 6
101 A(KLQ)=1+FIX(RND*10)
102 NEXT KLQ
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 6
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*6)
145 IF B=5 OR B=6 GOTO 164
159 REM   FOR IPP=1 TO FIX(1+RND*3)
160 REM   B=1+FIX(RND*6)
161 IF JJJJ<-31000 THEN  162 ELSE 164
162 IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE       X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 6
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B)  -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM  IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
181 IF  (.001*X(1)*X(2)*X(3)*X(4)*X(5)  )<1E-08 THEN 1670
184 X(6)=(2.07            )/  (    +.001*X(1)*X(2)*X(3)*X(4)*X(5)   )
194 REM  X(7)=-2.07+.001*X(1)*X(2)*X(3)*X(4)*X(5)*X(6)
195 X(8)=-.00062*X(1)*X(4)*X(5)^2 *(  X(1)+X(2)+X(3)     )    -.00058*X(2)*X(3)      *X(6)^2    *(X(1)+1.57*X(2)+X(4)    )       +1.2
211 FOR J44=1 TO 6
212 IF X(J44)<0 THEN 1670
214 NEXT J44
221 FOR J44=7 TO 8
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(7)        +P(8)
569  P=-.0204*X(1)*X(4)*(X(1)+X(2)+X(3)     )    -.0187*X(2)*X(3)*(X(1)+1.57*X(2)+X(4)    )      -.0607*X(1)*X(4)*X(5)^2 *(X(1)+X(2)+X(3)  ) -.0437*X(2)*X(3)*X(6)^2  *(X(1)+1.57*X(2)+X(4)      )   +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 8
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-131.06 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1908 PRINT A(7),A(8)
1989 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS.  The complete output through JJJJ=-31996 is shown below.  What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.    

 5   4   9
 10   .948811616411991   1.212042399564436
 0    4.492791434306553D-02
-131.0462978344562   -32000

 5   4   9
 10   .9488114244663351  1.212042644762025
 0    4.492791434306553D-02
-131.0462978344639   -31997

 5   4   9
 10   .9488116530299334   1.212042352787503
 0    4.492792595201128D-02
-131.0462978344565   -31996

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11,  the wall-clock time for obtaining the output shown above was 4 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
 
References

[1] Dana H. Ballard, C. O Jelinek, R. Schinzinger (1974), An Algorithm for the Solution of Constrained Generalised Polynomial Programming Problems.  The Computer Journal, Volume 17, Number 3, pp. 261-266.

[2] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition.  Brooks/Cole 2011.

[3] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Usink a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.

[4] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[5] K. Deb, M. Goyal (1996), A Combined Genetic Adaptive Search (GeneAS) for Engineering Design.  Computer Science and Informatics, 26 (4), pp. 30-45.

[6] R. J. Duffin, E. L. Peterson, C. Zener, Geometric Programming Theory and Applications.  John Wiley and Sons, 1967.

[7] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs.  Mathematical Programming, Vol. 36:307-339, 1986.

[8] F. Erbatur,  O. Hasancebi, I. Tutuncu, H. Kilic (2000) Optimal Design of Planar and Spcae Structures with Genetic Algorithms.  Computers and Structures, 75 (2), 209-224.

[9] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[10] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[11] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[12] Christodoulos A. Floudas, Nonlinear and Mixed-Integer Optimization.  Oxford University Press, 1995.

[13] C. A. Floudas, Deterministic Global Optimization.  Kluwer Academic Publishers, 2000.

[14] Amos Gilat and Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB.  John Wiley and Sons, Inc. 2008.

[15] Donald Greenspan, Vincenzo Casulli, Numerical Analysis for Applied Mathematics, Science, and Enginerring.  Addison-Wesley Publishing Company, 1988.

[16] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Springer-Verlag 1981.

[17] Majid Jaberipour, Esmaile Khorram (2010), Two Improved Hormony Search Algorithms for Solving Engineering Optimization Problems.  Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3316-3331.

[18] M. Jaberipour, E. Khorram (2010), Solving the Sum-of-Ratios Problems by a Harmony Search Algorithm.  Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[19] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed Discrete Engineering Optimization Problems.  Engineering Optimization, 05/2011, 43:507-523.

[20] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[21] B. K. Kannen , S. N. Kramer (1994), An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design.  Journal of Mechanical Design, 116, pp. 405-411.

[22] R. Baker Kearfott, Some Tests of Generalized Bisection.  ACM Transactions on Mathematical Software, Vol.13, No. 3, September 1987, pages 197-220.

[23] Esmaile Khorram, Majid Jaberipour, Harmony Search Algorithm for Solving Combined Heat and Power Economic Dispatch Problems.  Energy Conversion and Management 52 (2011) 1550-1554.

[24] G. R. Kocis, I. E. Grossmann, Relaxation Strategy for the Structural Optimization of Process Flow Sheets.  Ind. Eng. Chem. Res., 26 (9):1869 (1987).

[25} Sonia Krzyworzcka, Extension of the Lanczos and CGS Methods to Systems of Nonlinear Equations.  Journal of Computational and Aplied Mathematics 69 (1996) 181-190.

[26] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[27] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[28] Ya-Zhang Luo, Guo-Jing Tang, Li-Ni Zhou, Hybrid Approach for Solving Systems of Nonlinear Equations Using Chaos Optimization and Quasi-Newton Method.  Applied Soft Computing 8 (2008) 1068-1063.

[29] C. D. Maranas, C. A. Floudas, Finding All Solutions of Nonlinearly Constrained Systems of Equations.  Journal of Global Optimization, 7(2):143-182, 1995.

[30] Harry M. Markowitz and Alan S. Mann.  On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[31] Keith Meintjes and Alexander P. Morgan, Chemical Equilibrium Systems as Numerical Test Problems.  ACM Transactions on Mathematical Software, Vol. 16, No. 2, June 1990, Pages 143-151.

[32] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[33] Yuanbin Mo, Hetong Liu, Qin Wang,Conjugate Direction Particle Swarm Optimization Solving Systems of Nonlinear Equations. Computers and Mathematics with Applications 57 (2009) 1877-1882.

[34] Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, An Introduction to Interval Analysis.  Cambridge University Press, 2009.

[35] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition.  Cambridge University Press, 2000.

[36] Singiresu S. Rao, Ying Xiong (2005), A Hybrid Genetic Algorithm for Mixed-Discrete Design Optimization.  Transaction of the ASME, Vol. 127, November 2005, pp. 1100-1112.

[37] John R. Rice, Numerical Methods, Software, and Analysis, Second Edition.  Academic Press, 1993.

[38] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[39] E. Sandgren (1990), Nonlinear Integer and Discrete Programming in Mechanical Design Optimization.  Journal of Mechanical Design, 112, pp. 223-229.

[40] Jung-Fa Tsai (2010), Global Optimization for Signomial Discrete Programming Problems in Engineering Design.  Engineering Optimization, 42:9, pp. 833-843.

[41] L.-W. Tsai, A. P. Morgan, Solving the Kinematics of the Most General Six- and Five-Degree-of Freedom Manipulators by Continuation Methods.  Journal of Mechanisms, Transmissions, and Automation in Design, June 1985, Vol. 107, pp. 189-200.

[42] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[43] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[44] Jsun Yui Wong (2011, May 15).  The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/

[45] Jsun Yui Wong (2011, May 5).  The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/

[46] Jsun Yui Wong (2011, July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[47] Jsun Yui Wong (2012, April 10).  The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design.  Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/

[48] Jsun Yui Wong (2012, April 21).  The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/

[49] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[50] Jsun Yui Wong (2013, January 14).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition.   Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/

[51] Jsun Yui Wong (2013, January 24).  The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables.   Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/