Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Shao and Ehrgott [29, p. 723, Example 5.2]:

Minimize (5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4)

subject to

(X(1) - 2) ^ 2 + (X(2) - 2) ^ 2<=1,

0<=X(1) <= 3,

0<=X(2) <= 3.

One notes the big 2 above; in the preceding paper, it is 3.

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 * 3

75 A(2) = RND * 3

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) > 3 THEN 1670

211 IF X(2) < 0 THEN 1670

213 IF X(2) > 3 THEN 1670

311 X(3) = 1 - (X(1) - 2) ^ 2 - (X(2) - 2) ^ 2

333 FOR J44 = 3 TO 3

336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

378 POBA = -(5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4) + 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 [40]. The complete output through JJJJ = -31999.98 is shown below:

2.109949904817913 2.993937130019061 0

-76987.16781058605 -32000

2.104782367447188 2.994495176193511 0

-76969.52690960531 -31999.99

2.097450896297014 2.995240334196171 0

-76958.16722934964 -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 [40], 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."

**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 Sinulation 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. 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.

[29] Lizhen Shao, Matthias Ehrgott (2014). An objective space cut and bound algorithm for convex multiplicative programmes.* Journal of Global Optimization* (2014) 58:711-728.

[30] 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.

[31] 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.

[32] 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.

[33] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).

[34] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[35] 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

[36] 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.

[37] 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.

[38] 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.

[39] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.

[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[41] 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.

[42] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

## Comments

You can follow this conversation by subscribing to the comment feed for this post.