January 21, 2018 by 123writereadmath

Solving for Integer Solutions of Integer Nonlinear Programming Problems

Jsun Yui Wong

I. With Constraint (X(1) – 2) ^ 2 + (X(2) – 2) ^ 2<=1

The computer program listed below seeks to solve the following problem:

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) <= 11111,

0<=X(2) <= 11111,

X(1) and X(2) are integer variables.

The problem above is based on Shao and Ehrgott’s [29, p. 723] Example 5.2, which is as follows:

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.

X(3) below is an added slack variable.

One notes line 72 and line 75, which are 72 A(1) = RND * 11111 and 75 A(2) = RND * 11111, respectively.

One also notes line 201 through 213, which are as follows:
201 IF X(1) < 0 THEN 1670
203 IF X(1) > 11111 THEN 1670
211 IF X(2) < 0 THEN 1670
213 IF X(2) > 11111 THEN 1670.

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 * 11111

75 A(2) = RND * 11111

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

193 X(1) = INT(X(1))

194 X(2) = INT(X(2))

201 IF X(1) < 0 THEN 1670
203 IF X(1) > 11111 THEN 1670

211 IF X(2) < 0 THEN 1670
213 IF X(2) > 11111 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 = -31998.35000000026 is shown below:

2 3 0 -78246 -31999.02000000016

2 3 0 -78246 -31998.94000000017

2 3 0 -78246 -31998.89000000018

2 3 0 -78246 -31998.7700000002

2 3 0 -78246 -31998.35000000026

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= -31998.35000000026 was about 2 seconds, not including the time for “Creating .EXE file.”

II. With Constraint (X(1) – 2) ^ 2 + (X(2) – 3) ^ 2<=1

The computer program listed below seeks to solve the following problem:

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) – 3) ^ 2<=1,

0<=X(1) <= 11111,

0<=X(2) <= 11111,

X(1) and X(2) are integer variables.

The problem above is based on Shen, Zhang, and Wang’s [32, p. 12 of 16] Example 5, which is as follows:

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) – 3) ^ 2<=1,

0<=X(1) <= 3,

0<=X(2) <= 3.

X(3) below is an added slack variable.

One notes line 72 and line 75, which are 72 A(1) = RND * 11111 and 75 A(2) = RND * 11111, respectively.

One also notes line 201 through 213, which are as follows:
201 IF X(1) < 0 THEN 1670
203 IF X(1) > 11111 THEN 1670
211 IF X(2) < 0 THEN 1670
213 IF X(2) > 11111 THEN 1670.

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 * 11111

75 A(2) = RND * 11111

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

193 X(1) = INT(X(1))

194 X(2) = INT(X(2))

201 IF X(1) < 0 THEN 1670
203 IF X(1) > 11111 THEN 1670

211 IF X(2) < 0 THEN 1670
213 IF X(2) > 11111 THEN 1670

311 X(3) = 1 – (X(1) – 2) ^ 2 – (X(2) – 3) ^ 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 = -31998.54000000023 is shown below:

2 4 0 0 -31999.24000000012

2 4 0 0 -31999.12000000014

2 4 0 0 -31999.08000000015

2 4 0 0 -31998.90000000018

2 4 0 0 -31998.7900000002

2 4 0 0 -31998.64000000022

2 4 0 0 -31998.54000000023

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= -31998.54000000023 was about 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. ScienApplied 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.
[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.

January 21, 2018 by 123writereadmath

Integer Solution of an Integer, Nonlinear, and Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following problem:

Minimize ((-X(1) + 2 * X(2) + 2) / (3 * X(1) – 4 * X(2) + 5)) * ((4 * X(1) – 3 * X(2) + 4) / (-2 * X(1) + X(2) + 3))

subject to

X(1) + X(2)<=1.5,

X(1) <= X(2),

0<=X(1) <= 1,

0<=X(2) <= 1,

0<=X(1) <= 11111,

0<=X(2) <= 11111,

X(1) and X(2) are integer variables.

The problem above is based on Shen, Zhang, and Wang’s [32, p. 11 of 16] Example 2, which is as follows:

Minimize ((-X(1) + 2 * X(2) + 2) / (3 * X(1) – 4 * X(2) + 5)) * ((4 * X(1) – 3 * X(2) + 4) / (-2 * X(1) + X(2) + 3))

subject to

X(1) + X(2)<=1.5,

X(1) <= X(2),

0<=X(1) <= 1,

0<=X(2) <= 1.

X(3) and X(4) below are added slack variables.

One notes line 72 and line 75, which are 72 A(1) = RND * 11111 and 75 A(2) = RND * 11111, respectively.

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 * 11111

75 A(2) = RND * 11111

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

193 X(1) = INT(X(1))
194 X(2) = INT(X(2))

201 IF X(1) < 0 THEN 1670
203 IF X(1) > 11111 THEN 1670

211 IF X(2) < 0 THEN 1670
213 IF X(2) > 11111 THEN 1670

311 X(3) = 1.5 – X(1) – X(2)
313 X(4) = 0 – X(1) + X(2)

333 FOR J44 = 3 TO 4

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

339 NEXT J44
388 POBA = -((-X(1) + 2 * X(2) + 2) / (3 * X(1) – 4 * X(2) + 5)) * ((4 * X(1) – 3 * X(2) + 4) / (-2 * X(1) + X(2) + 3)) + 1000000 * (X(3) + X(4))

466 P = POBA

1111 IF P <= M THEN 1670

1452 M = P
1454 FOR KLX = 1 TO 4

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), X(4), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [40]. The complete output through JJJJ = -31999.95000000001 is shown below:

0 0 0 0 -.5333333333333333
-32000

0 0 0 0 -.5333333333333333
-31999.99

0 0 0 0 -.5333333333333333
-31999.98

0 0 0 0 -.5333333333333333
-31999.97000000001

0 0 0 0 -.5333333333333333
-31999.96000000001

0 0 0 0 -.5333333333333333
-31999.95000000001

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= -31999.95000000001 was about 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 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.
[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.

January 19, 2018 by 123writereadmath

Solving a Nonlinear Programming Problem Involving a Product of 3 Nonlinear Functions

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.

[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.

January 19, 2018 by 123writereadmath

Solving a Fractional Programming Problem Involving a Sum of 2 Ratios

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Shen, Zhang, and Wang [31, p. 11 of 16, Example 1]:

Minimize ((-X(1) + 2 * X(2) + 2) / (3 * X(1) – 4 * X(2) + 5)) + ((4 * X(1) – 3 * X(2) + 4) / (-2 * X(1) + X(2) + 3))

subject to

X(1) + X(2)<=1.5,

X(1) <= X(2),

0<=X(1) <= 1,

.0<=X(2) <= 1.

X(3) and X(4) below are added slack variables.

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

75 A(2) = RND

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

193 REM X(1) = 2.0814
194 REM X(2) = 2.9963

201 IF X(1) < 0 THEN 1670
203 IF X(1) > 1 THEN 1670

211 IF X(2) < 0 THEN 1670
213 IF X(2) > 1 THEN 1670

311 X(3) = 1.5 – X(1) – X(2)
313 X(4) = 0 – X(1) + X(2)

333 FOR J44 = 3 TO 4

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

339 NEXT J44

388 POBA = -(((-X(1) + 2 * X(2) + 2) / (3 * X(1) – 4 * X(2) + 5)) + ((4 * X(1) – 3 * X(2) + 4) / (-2 * X(1) + X(2) + 3))) + 1000000 * (X(3) + X(4))

466 P = POBA

1111 IF P <= M THEN 1670

1452 M = P
1454 FOR KLX = 1 TO 4

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), X(4), 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:

1.585023957168358D-17 .2839473937478193 0
0 -1.62318335773863 -32000

1.218197441737778D-16 .283947398694873 0
0 -1.62318335773863 -31999.99

1.234716735768767D-18 .283947391945479 0
0 -1.62318335773863 -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 1].

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. 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.

January 15, 2018 by 123writereadmath

Solving a Nonlinear Fractional Programming Problem Involving 1 Nonlinear Numerator, 2 Nonlinear Denominators and 3 Nonlinear Constraints

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Shen, Ma, and Chen [30, p. 453, Example 6].

Maximize -(-1.35) * ((X(1) ^ 2 * X(2) ^ .5 – 2 * X(1) * X(2) ^ -1 + X(2) ^ 2 – 2.8 * X(1) ^ -1 * X(2) + 7.5) / (X(1) * X(2) ^ 1.5 + 1)) – (12.99) * ((X(2) + .1) / (X(1) ^ 2 * X(2) ^ -1 – 3 * X(1) ^ -1 + 2 * X(1) * X(2) ^ 2 – 9 * X(2) ^ -1 + 12))

subject to

2 * X(1) ^ -1 + X(1) * X(2)<=4,

X(1) +3 * X(1) ^ -1 * X(2)<=5,

X(1) ^ 2 – 3 * X(2) ^ 3<=2,

1<=X(1) <= 3,

1<=X(2) <= 3.

X(3) through X(5) below are added slack variables.

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) = 1 + RND * 2

75 A(2) = 1 + RND * 2

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) < 1 THEN 1670
203 IF X(1) > 3 THEN 1670

211 IF X(2) < 1 THEN 1670
213 IF X(2) > 3 THEN 1670

311 X(3) = 4 – 2 * X(1) ^ -1 – X(1) * X(2)

313 X(4) = 5 – X(1) – 3 * X(1) ^ -1 * X(2)

315 X(5) = 2 – X(1) ^ 2 + 3 * X(2) ^ 3

333 FOR J44 = 3 TO 5
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

378 POBA = -(-1.35) * ((X(1) ^ 2 * X(2) ^ .5 – 2 * X(1) * X(2) ^ -1 + X(2) ^ 2 – 2.8 * X(1) ^ -1 * X(2) + 7.5) / (X(1) * X(2) ^ 1.5 + 1)) – (12.99) * ((X(2) + .1) / (X(1) ^ 2 * X(2) ^ -1 – 3 * X(1) ^ -1 + 2 * X(1) * X(2) ^ 2 – 9 * X(2) ^ -1 + 12)) + 1000000 * (X(3) + X(4) + X(5))

466 P = POBA

1111 IF P <= M THEN 1670

1452 M = P
1454 FOR KLX = 1 TO 5

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -11111 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [38]. The complete output through JJJJ = -31999.98 is shown below:

2.698672512963932 1.207578072004017 0
0 0 2.332207556891017 -32000

2.698682666462486 1.20758224763645 0
0 0 2.332213594166706 -31999.99

2.698690059295716 1.20758528793531 0
0 0 2.332217989915511 -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 [38], 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, Ma, and Chen [30, p.452, Table 1].

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. 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.
[29] 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] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[32] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[33] 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
[34] 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.
[35] 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.
[36] 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.
[37] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[38] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[39] 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.
[40] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

January 13, 2018 by 123writereadmath

Solving Another Nonlinear Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Benson [3, pp. 361-362, Example 3], which is Example 2 in Dur et al. [9, pp. 36-37].

Maximize ((-X(1) ^ 2 + 16 * X(1) – X(2) ^ 2 + 16 * X(2) – X(3) ^ 2 + 16 * X(3) – X(4) ^ 2 + 16 * X(4) – 214) / (2 * X(1) – X(2) – X(3) + X(4) + 2)) + ((-X(1) ^ 2 + 16 * X(1) – 2 * X(2) ^ 2 + 20 * X(2) – 3 * X(3) ^ 2 + 60 * X(3) – 4 * X(4) ^ 2 + 56 * X(4) –
586) / (-X(1) + X(2) + X(3) – X(4) + 10)) + ((-X(1) ^ 2 + 20 * X(1) – X(2) ^ 2 + 20 * X(2) – X(3) ^ 2 + 20 * X(3) – X(4) ^ 2 + 20 * X(4) – 324) / (X(1) ^ 2 – 4 * X(4)))

subject to

X(1) + X(2) + X(3) + X(4)<=34,

6<= X(1) <=10,

4<= X(2) <=6,

8<= X(3) <=12,

6<= X(4) <=8.

X(5) below is an added slack variable.

Whereas line 193, for example, of the preceding paper is 193 X(1) = INT(X(1)), here line 193 is 193 REM X(1) = INT(X(1)).

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) = 6 + RND * 4

75 A(2) = 4 + RND * 2

77 A(3) = 8 + RND * 4

79 A(4) = 6 + RND * 2

128 FOR I = 1 TO 500

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 4)

183 r = (1 – RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

193 REM X(1) = INT(X(1))

194 REM X(2) = INT(X(2))

195 REM X(3) = INT(X(3))

199 REM X(4) = INT(X(4))

201 IF X(1) < 6 THEN 1670
203 IF X(1) > 10 THEN 1670

211 IF X(2) < 4 THEN 1670
213 IF X(2) > 6 THEN 1670

221 IF X(3) < 8 THEN 1670
223 IF X(3) > 12 THEN 1670

231 IF X(4) < 6 THEN 1670
233 IF X(4) > 8 THEN 1670

311 X(5) = 34 – X(1) – X(2) – X(3) – X(4)

333 FOR J44 = 5 TO 5
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

368 POBA = ((-X(1) ^ 2 + 16 * X(1) – X(2) ^ 2 + 16 * X(2) – X(3) ^ 2 + 16 * X(3) – X(4) ^ 2 + 16 * X(4) – 214) / (2 * X(1) – X(2) – X(3) + X(4) + 2)) + ((-X(1) ^ 2 + 16 * X(1) – 2 * X(2) ^ 2 + 20 * X(2) – 3 * X(3) ^ 2 + 60 * X(3) – 4 * X(4) ^ 2 + 56 * X(4) – 586) / (-X(1) + X(2) + X(3) – X(4) + 10)) + ((-X(1) ^ 2 + 20 * X(1) – X(2) ^ 2 + 20 * X(2) – X(3) ^ 2 + 20 * X(3) – X(4) ^ 2 + 20 * X(4) – 324) / (X(1) ^ 2 – 4 * X(4))) + 1000000 * (X(5))

466 P = POBA

1111 IF P <= M THEN 1670

1452 M = P
1454 FOR KLX = 1 TO 5

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -11111 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5)
1912 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [38]. The complete output through JJJJ = -31999.98 is shown below:

6.000000016539983 5.999999997706385 10.05502047670345
7.99999999999587 0
16.16857695236145 -32000

6.000000002306647 5.999999998909268 10.0550212753023
7.999999994885505 0
16.1685773138853 -31999.99

6.00000000030631 5.99999999987464 10.0550213812697
7.999999996864687 0
16.16857739301814 -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 [38], the wall-clock time for obtaining the output through
JJJJ= -31999.98 was 2 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above to the results in Benson [2, p.362].

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.
[29] 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] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[32] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[33] 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
[34] 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.
[35] 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.
[36] 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.
[37] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[38] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[39] 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.
[40] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

January 13, 2018 by 123writereadmath

Solving Another Integer, Nonlinear, and Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve Example 3 in Benson [3, pp. 361-362], which is Example 2 in Dur et al. [9, pp. 36-37], plus the restriction that the four variables are integer variables. Thus the present problem is as follows:.

586) / (-X(1) + X(2) + X(3) – X(4) + 10)) + ((-X(1) ^ 2 + 20 * X(1) – X(2) ^ 2 + 20 * X(2) – X(3) ^ 2 + 20 * X(3) – X(4) ^ 2 + 20 * X(4) – 324) / (X(1) ^ 2 – 4 * X(4)))

subject to

X(1) + X(2) + X(3) + X(4)<=34,

6<= X(1) <=10,

4<= X(2) <=6,

8<= X(3) <=12,

6<= X(4) <=8,

X(1) through X(4) are integer variables.

X(5) below is an added slack variable.

One notes lines 193, 194, 195, and 199.

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) = 6 + RND * 4

75 A(2) = 4 + RND * 2

77 A(3) = 8 + RND * 4

79 A(4) = 6 + RND * 2

128 FOR I = 1 TO 500

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 4)

183 r = (1 – RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

193 X(1) = INT(X(1))

194 X(2) = INT(X(2))

195 X(3) = INT(X(3))

199 X(4) = INT(X(4))

201 IF X(1) < 6 THEN 1670
203 IF X(1) > 10 THEN 1670

211 IF X(2) < 4 THEN 1670
213 IF X(2) > 6 THEN 1670

221 IF X(3) < 8 THEN 1670
223 IF X(3) > 12 THEN 1670

231 IF X(4) < 6 THEN 1670
233 IF X(4) > 8 THEN 1670

311 X(5) = 34 – X(1) – X(2) – X(3) – X(4)

333 FOR J44 = 5 TO 5
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

466 P = POBA

1111 IF P <= M THEN 1670

1452 M = P
1454 FOR KLX = 1 TO 5

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -11111 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5)
1912 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [38]. The complete output through JJJJ = -31999.95000000001 is shown below:

6 6 10 8 0
16.16666666666667 -32000

6 6 10 8 0
16.16666666666667 -31999.99

6 6 10 8 0
16.16666666666667 -31999.98

6 6 10 8 0
16.16666666666667 -31999.97000000001

6 6 10 8 0
16.16666666666667 -31999.96000000001

6 6 10 8 0
16.16666666666667 -31999.95000000001

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 [38], the wall-clock time for obtaining the output through
JJJJ= -31999.95000000001 was 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

[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] 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] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[32] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[33] 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
[34] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. 178 (2007) pp. 10-19.European Journal of Operational Research.
[35] 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.
[36] 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.
[37] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[38] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[39] 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.
[40] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

January 13, 2018 by 123writereadmath

Solving a Mixed Integer, Nonlinear, and Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve Example 1 in Jaberipour and Khorram [13, p. 738, Example 1] plus the restriction that X(3) is an integer variable. Thus the present problem is as follows:.

Maximize ((13 * X(1) + 13 * X(2) + 13) / (37 * X(1) + 73 * X(2) + 13)) ^ -1.4 * ((64 * X(1) – 18 * X(2) + 39) / (13 * X(1) + 26 * X(2) + 13)) ^ 1.2 – ((X(1) + 2 * X(2) + 5 * X(3) + 50) / (X(1) + 5 * X(2) + 5 * X(3) + 50)) ^ .5 * ((X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(2) + 4 * X(3) + 50)) ^ -2

subject to

2*X(1) + X(2)+5* X(3)<=10,

5*X(1) -3* X(2) =3,

1.5 <= X(1)<=3, X(2), X(3) free,

X(1) and X(2) are continuous variables,

X(3) is an integer variable.

X(4) below is an added slack variable.

The following computer program is similar to the computer programs of the two preceding papers. One notes line 194, which is 194 X(3) = INT(X(3)) .

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
70 FOR J44 = 1 TO 3

72 A(J44) = RND * 10

73 NEXT J44

128 FOR I = 1 TO 500

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

181 J = 1 + FIX(RND * 3)

183 r = (1 – RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP
193 REM X(2) = INT(X(2))

194 X(3) = INT(X(3))

196 REM X(1) = INT((3 + 3 * X(2)) / 5)

198 X(1) = ((3 + 3 * X(2)) / 5)
201 IF X(1) < 1.5 THEN 1670
203 IF X(1) > 3 THEN 1670

205 REM FOR J44 = 1 TO 3

206 REM IF X(J44) < 0 THEN 1670

208 REM NEXT J44

209 REM IF X(1) = ((3 + 3 * X(2)) / 5) THEN 311 ELSE GOTO 1670

311 X(4) = 10 – 2 * X(1) – X(2) – 5 * X(3)

333 FOR J44 = 4 TO 4
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

368 POBA = ((13 * X(1) + 13 * X(2) + 13) / (37 * X(1) + 73 * X(2) + 13)) ^ -1.4 * ((64 * X(1) – 18 * X(2) + 39) / (13 * X(1) + 26 * X(2) + 13)) ^ 1.2 – ((X(1) + 2 * X(2) + 5 * X(3) + 50) / (X(1) + 5 * X(2) + 5 * X(3) + 50)) ^ .5 * ((X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(2) + 4 * X(3) + 50)) ^ -2 + 1000000 * (X(4))

466 P = POBA

1111 IF P <= M THEN 1670

1452 M = P
1454 FOR KLX = 1 TO 4

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < 8.12 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4)
1912 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [36]. The complete output through JJJJ = -31999.74000000001 is shown below:

1.499999940521272 1.499999900868787 1
0
8.12032938241665 -31999.85000000002

1.499999943870452 1.499999906450753 1
0
8.120329374322648 -31999.84000000003

1.499999940538614 1.49999990089769 1
0
8.12032938237474 -31999.74000000004

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 [36], the wall-clock time for obtaining the output through
JJJJ= -31999.74000000001 was 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] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[4] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[5] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[6] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] 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.
[10] 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.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[16] 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.
[17] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[18] 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.
[19] 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.
[20] 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.
[21] 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/
[22] 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.
[23] 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…/
[24] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[25] 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.
[26] 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.
[27] 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.
[28] 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.
[29] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[30] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[31] 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
[32] 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.
[33] 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.
[34] 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.
[35] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[36] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[37] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[38] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[39] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[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.

January 11, 2018 by 123writereadmath

Solving Another Nonlinear and Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Jaberipour and Khorram [13, p. 738, Example 1].

subject to

2*X(1) + X(2)+5* X(3)<=10,

5*X(1) -3* X(2) =3,

1.5 <= X(1)<=3, X(2), X(3) free.

X(4) and X(5) below are added slack variables.

The following computer program is similar to the computer program of the preceding paper. One notes the following line 193, line 194, line 196, line 198, and line 209.

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
70 FOR J44 = 1 TO 3

72 A(J44) = RND * 10

73 NEXT J44

128 FOR I = 1 TO 500

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

181 J = 1 + FIX(RND * 3)

183 r = (1 – RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP
193 REM X(2) = INT(X(2))

194 REM X(3) = INT(X(3))

196 REM X(1) = INT((3 + 3 * X(2)) / 5)

198 X(1) = ((3 + 3 * X(2)) / 5)
201 IF X(1) < 1.5 THEN 1670
203 IF X(1) > 3 THEN 1670

205 REM FOR J44 = 1 TO 3

206 REM IF X(J44) < 0 THEN 1670

208 REM NEXT J44

209 REM IF X(1) = ((3 + 3 * X(2)) / 5) THEN 311 ELSE GOTO 1670

311 X(4) = 10 – 2 * X(1) – X(2) – 5 * X(3)

333 FOR J44 = 4 TO 4
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

466 P = POBA

1111 IF P <= M THEN 1670

1452 M = P
1454 FOR KLX = 1 TO 4

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -11111 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4)
1912 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [36]. The complete output through JJJJ = -31999.95000000001 is shown below:

1.499999940396006 1.49999990066001 1.100000043671674
0
8.120723176456364 -32000

1.499999940414173 1.499999900690288 1.100000042436616
0
8.120723176407633 -31999.99

1.499999940420794 1.499999900701324 1.100000043277437
0
8.120723176394925 -31999.98

1.499999940403065 1.499999900671775 1.100000042872618
0
8.120723176436181 -31999.97000000001

1.499999940569063 1.499999900948438 1.100000036882215
0
8.120723176011648 -31999.95000000001

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 [36], the wall-clock time for obtaining the output through JJJJ= -31999.95000000001 was one or two seconds, not including the time for “Creating .EXE file.” One can compare the computational results above to the results in Jaberipour and Khorram [13, pp.739-740].

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] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[4] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[5] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[6] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] 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.
[10] 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.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[16] 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.
[17] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[18] 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.
[19] 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.
[20] 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.
[21] 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/
[22] 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.
[23] 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…/
[24] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[25] 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.
[26] 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.
[27] 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.
[28] 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.
[29] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[30] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[31] 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
[32] 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.
[33] 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.
[34] 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.
[35] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[36] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[37] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[38] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[39] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[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

Nonlinear Integer Programming

Month: January 2018

Solving for Integer Solutions of Integer Nonlinear Programming Problems

Integer Solution of an Integer, Nonlinear, and Fractional Programming Problem

Related Integer Solution of the Preceding Problem

Solving a Nonlinear Programming Problem Involving a Product of 3 Nonlinear Functions

Solving a Fractional Programming Problem Involving a Sum of 2 Ratios

Solving a Nonlinear Fractional Programming Problem Involving 1 Nonlinear Numerator, 2 Nonlinear Denominators and 3 Nonlinear Constraints

Solving Another Nonlinear Fractional Programming Problem

Solving Another Integer, Nonlinear, and Fractional Programming Problem

Solving a Mixed Integer, Nonlinear, and Fractional Programming Problem

Solving Another Nonlinear and Fractional Programming Problem