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 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/
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