Jsun Yui Wong
The computer program listed below seeks to solve the following problem in Shen, Zhang, and Wang [31, p. 12 of 16, Example 4]:
Minimize ((X(1) – 1) ^ 2 + (X(2) – 1) ^ 2 + 1) * ((X(1) – 2) ^ 2 + (X(2) – 3) ^ 2 + 1) * ((X(1) – 4) ^ 2 + (X(2) – 2) ^ 2 + 1)
subject to
X(1) + 2*X(2)<=10,
0<=X(1) <= 10,
.0<=X(2) <= 4.
X(3) below is an added slack variable.
0 DEFDBL A-Z
2 DEFINT K
3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000 STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
72 A(1) = RND * 10
75 A(2) = RND * 4
128 FOR I = 1 TO 2000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))
181 J = 1 + FIX(RND * 2)
183 r = (1 – RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r
191 NEXT IPP
201 IF X(1) < 0 THEN 1670
203 IF X(1) > 10 THEN 1670
211 IF X(2) < 0 THEN 1670
213 IF X(2) > 4 THEN 1670
311 X(3) = 10 – X(1) – 2 * X(2)
333 FOR J44 = 3 TO 3
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
339 NEXT J44
388 POBA = -((X(1) – 1) ^ 2 + (X(2) – 1) ^ 2 + 1) * ((X(1) – 2) ^ 2 + (X(2) – 3) ^ 2 + 1) * ((X(1) – 4) ^ 2 + (X(2) – 2) ^ 2 + 1) + 1000000 * (X(3))
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 3
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -111111 THEN 1999
1900 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [39]. The complete output through JJJJ = -31999.98 is shown below:
2.188967691603656 2.433096908147784 0
-27.08443533712395 -32000
2.188967706327333 2.433096919790813 0
-27.08443533712396 -31999.99
2.188967700758801 2.433096913634957 0
-27.08443533712395 -31999.98
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [39], the wall-clock time for obtaining the output through
JJJJ= -31999.98 was 1 or 2 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above to the results in Shen, Zhang, and Wang [31, p. 10 of 16, Table 1, Example 4].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Mohamed Abdel-Baset, Ibrahim M. Hezam (2015). An Improved flower pollination algorithm for ratios optimization problems. Applied Mathematics and Information Sciences Letters: An International Journal, 3, No. 2, 83-91 (2015). http://dx.doi.org/10.12785/amisl/030206.
[2] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[3] Harold P. Benson (2002). Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22: 343-364 (2002)
[4] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[5] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[6] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[7] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[8] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[9] Mirjam Dur, Charoenchai Khompatraporn, Zelda B. Zabinsky (2007). Solving fractional problems with dynamic multistart improving hit-and-run. Annals of Operations Research (2007) 156:25-44.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[11] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[12] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[13] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[14] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.
[15] Majid Jaberipour, Esmaile Khorram (2010). Solving the sum-of-ratios problems by a harmony search algorithm. Journal of Computational and Applied Mathematics 234 (2010) 733-742.
[16] Yun-Chol Jong (2012). An efficient global optimization algorithm for nonlinear sum-of-ratios problem. http://www.optimization-online.org/DB_FILE/2012/08/3586.pdf.
[17] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[18] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[19] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[20] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[21] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[22] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[23] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[24] 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.
[25] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[26] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[27] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[28] C. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[290] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei (2009). A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[30] Pei Ping Shen, Yuan Ma, Yongqiang Chen (2011). Global optimization for the generalized polynomial for the sum of ratios problem. Journal of Global Optimization (2011) 50:439-455.
[31] Peiping Shen, Tongli Zhang, Chunfeng Wang (2017). Solving a class of generalized fractional programming problems using the feasibility of linear programs.
Journal of Inequalities and Applications (2017) 207:147.
[32] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[33] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[34] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[35] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[36] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[37] Chun-Feng Wang, Xin-Yue Chu (2017). A new branch and bound method for solving sum of linear ratios problem. IAENG International Journal of Applied Mathematics 47:3, IJAM_47_3_06.
[38] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[39] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[40] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[41] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.
computerprogramsandresults 