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 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.
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