Substance

Solving a Heat Exchanger Design Problem Using the Method of This Blog

Jsun Yui Wong

The computer program listed below seeks to solve the following heat exchanger design problem from Chootinan and Chen [4, p. 2280] and Yang and Gandomi [25, pp. 477-479]:

Minimize X(1) + X(2) + X(3)

subject to

.0025 * (X(4) + X(6))-1<=0,

.0025 * (X(5) + X(7) - X(4))-1<=0,

.01 * (X(8) - X(5))-1<=0,

833.33252 * X(4) + 100 * X(1) - X(1) * X(6)-83333.333<=0,

-1250 * X(4) + 1250 * X(5) + X(2) * X(4) - X(2) * X(7)<=0,

-2500 * X(5) - X(3) * X(8) + X(3) * X(5)+1250000<=0,

100<= X(1)<= 10000,

1000<= X(i)<= 10000, i=2, 3,

10 <= X(i)<= 1000, i=4, 5, 6, 7, 8.

X(10), X(13), and X(14) below are slack variables.

Hoping to get some domino effect, one can change some of the inequality constraints to equalility constraints, and if these changes are not satisfactory, one can make other similar changes For examples, see lines 197, 200, and 201, which are 197 X(4) = (0 + 1 - .0025 * X(6)) / .0025, 200 X(1) = (83333.333 - 833.33252 * X(4)) / (100 - X(6)), and
201 X(8) = (1 + .01 * X(5)) / .01.

 

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 FOR J44 = 4 TO 8

 

74 A(J44) = 10 + RND * 990

 

79 NEXT J44

86 A(1) = 100 + RND * 9900

 

87 A(2) = 1000 + RND * 9000

 

90 A(3) = 1000 + RND * 9000

 

128 FOR I = 1 TO 25000

 

129 FOR KKQQ = 1 TO 8

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 8)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r
191 NEXT IPP

197 X(4) = (0 + 1 - .0025 * X(6)) / .0025

200 X(1) = (83333.333 - 833.33252 * X(4)) / (100 - X(6))

201 X(8) = (1 + .01 * X(5)) / .01

203 X(10) = 1 - .0025 * (X(5) + X(7) - X(4))

 

211 X(13) = 1250 * X(4) - 1250 * X(5) - X(2) * X(4) + X(2) * X(7)

 

212 X(14) = -125D+04 + 2500 * X(5) + X(3) * X(8) - X(3) * X(5)

 

 

235 IF X(1) < 100 THEN 1670

236 IF X(1) > 10000 THEN 1670
245 IF X(2) < 1000 THEN 1670

246 IF X(2) > 10000 THEN 1670

247 IF X(3) < 1000 THEN 1670

248 IF X(3) > 10000 THEN 1670

 

249 FOR J99 = 4 TO 8

 

251 IF X(J99) < 10 THEN 1670

252 IF X(J99) > 1000 THEN 1670

253 NEXT J99

 

259 FOR J99 = 9 TO 14

 

 

269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

270 NEXT J99

 

357 POBA = -X(1) - X(2) - X(3) + 1000000 * (X(10) + X(13) + X(14))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 14

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -7051 THEN 1999

 

1900 PRINT A(1), A(2), A(3)

1903 PRINT A(4), A(5), A(6)
1906 PRINT A(7), A(8)

 

1910 PRINT A(10), A(13), A(14)

1912 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [22]. Each candidate solution through JJJJ = -31992.77000000116 is summarized below:

 

-7049.937967496634       -31999.13000000014

 

-7049.299892835455       -31993.54000000103

 

582.5426733687975          1353.967849952792          5112.751288378784
182.2873486444452          295.4899484648486          217.7126513555548
286.7974001795965          395.4899484648486
0          0          0
-7049.261811700373       -31992.77000000116

   

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. One can compare the solutions above with the solutions in Table XII of Yang and Gandomi [25, p. 478]. One notes line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [22], the wall-clock time for obtaining the output through
JJJJ= -31992.77000000116 was 13 minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .

[3] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.

[5] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.

[6] 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.

[7] 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.

[8] 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.

[9] 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.

[10] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.9

[11] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[12] 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/

[13] Ong Kok Meng, Ong Pauline, Sia Chee Kiong, H. A. Wahab, N. Jafferi. Application of Modified Flower Pollination Algorithm on Mechanical Engineering Design Problem. IOP Conf. Series: Materials Science and Engineering 165 (2017) 012032.

[14] 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.

[15] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[16] 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.

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

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

[19] 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

[20] 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.

[21] 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.

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

[23] 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/

[24] 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.

[25] 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.

Solving a Griewank Problem in 5000 Continuous Variables Using the Method of This Blog

Jsun Yui Wong

The computer program listed below works on the following Griewank problem based on a problem in Gandomi et al. [5, p. 93]. (Also see www.mathworld.wolfram.com/GriewankFunction/html.)

Minimize

                         5000                5000
1+(1 / 4000)*SIGMA X(i)^2-PI COS(X(i)/i^.5 )
                          i=1                    i=1

where -600<=X(i)<=600, i=1, 2, 3,..., 5000.

 

0 DEFDBL A-Z

2 DEFINT K

 

3 DIM B(5099), N(5099), A(5099), H(5099), L(5099), U(5099), X(5011), D(5011), P(5011), PS(5033)

 

12 FOR JJJJ = -32000 TO 32000 STEP .01

 

14 RANDOMIZE JJJJ
16 M = -1D+37

75 FOR J44 = 1 TO 5000

 

76 A(J44) = -600 + RND * 1200

 

79 NEXT J44

 

128 FOR I = 1 TO 60

 

 

129 FOR KKQQ = 1 TO 5000

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 5000)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

193 FOR J44 = 1 TO 5000

 

221 IF X(J44) < -600 THEN 1670

 

222 IF X(J44) > 600 THEN 1670

 

223 NEXT J44
241 SUMOF = 0
244 FOR J66 = 1 TO 5000

247 SUMOF = SUMOF + X(J66) ^ 2

249 NEXT J66

259 REM GOTO 332

292 PRODOF = 1

293 FOR J77 = 1 TO 5000

295 PRODOF = PRODOF * COS(X(J77) / J77 ^ .5)

 

296 NEXT J77

 

329 POBA = -1 - (1 / 4000) * (SUMOF) + (PRODOF)

332 REM POBA = -(SUMOF) ^ .5

 

466 p = POBA

1111 IF p <= M THEN 1670

 

1452 M = p
1454 FOR KLX = 1 TO 5000

 

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1499 REM PRINT A(1), A(1200), M, JJJJ

 

1557 GOTO 128

1670 NEXT I
1889 REM IF M < -.000001 THEN 1999

1900 PRINT A(1), A(2), A(4999)

1936 PRINT A(5000), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [20]. The complete output through JJJJ= -31999.99 is shown below:

-5.918688677179165D-09       -5.320081384432523D-08       1.253728752978195D-07
4.421595063858971D-08       -2.215124330319306D-02       -32000

-1.705762443078007D-09       2.445902572237596D-09       1.728866761139114D-08
5.598867908251804D-09       -1.245862137413956D-13       -31999.99

 Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. Only the A's of line 1900 and line 1936 are shown above. The candidate solution at JJJJ=-31999.99 is better than the one at JJJJ=-32000. Only the A's of line 1900 and line 1936 are shown above. M=0 is optimal; see Gandomi, Yang, Talatahari, and Alavi [5, p. 93].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [20], the wall-clock time for obtaining the output through JJJJ=-31999.99 was 47 minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .

[3] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.

[5] 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.

[6] 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.

[7] 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.

[8] 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.

[9] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.9

[10] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[11] 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/

[12] 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.

[13] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[14] 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.

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

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

[17] 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

[18] 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.

[19] 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.

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

[21] 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/

[22] 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.

 

Solving a Speed Reducer Design Problem Using the Method of This Blog, Second Edition

Jsun Yui Wong

The computer program listed below works on the mathematical formulation on page 9 of Ong Kok Meng et al. [12]. (Please see
www.iopscience.iop.org/article/10.1088/1757-899X/165/1/012032/pdf.)

The follwing X(8) through X(18) are slack variables. Whereas line 128 of the earlier edition is 128 FOR I = 1 TO 5000, here line 128 is 128 FOR I = 1 TO 2500.

One notes line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

 

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 FOR J44 = 1 TO 7

 

74 A(J44) = 1 + RND * 10

 

79 NEXT J44
86 A(2) = .7 + RND

 

88 A(3) = 17 + FIX(RND * 12)
128 FOR I = 1 TO 2500

 

129 FOR KKQQ = 1 TO 7

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 7)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r
191 NEXT IPP

197 X(3) = INT(X(3))

 

204 X(8) = 1 - 27 / (X(1) * X(2) ^ 2 * X(3))

205 X(9) = 1 - 397.5 / (X(1) * X(2) ^ 2 * X(3) ^ 2)
206 X(10) = 1 - 1.93 / (X(2) * X(3) * X(6) ^ 4)
207 X(11) = 1 - 1.93 / (X(2) * X(3) * X(6) ^ 4)

 

208 y1 = (745 * X(4) / (X(2) * X(3))) ^ 2

 

209 IF (y1 + 16.9D+06) < 0 THEN 1670

 

 

210 X(12) = 1 - ((1 / (110 * X(6) ^ 3)) * (y1 + 16.9D+06) ^ .5)

 

211 y2 = (745 * X(5) / (X(2) * X(3))) ^ 2

 

212 IF (y2 + 157.5D+06) < 0 THEN 1670

 

 

213 X(13) = 1 - ((1 / (85 * X(7) ^ 3)) * (y2 + 157.5D+06) ^ .5)

 

 

216 X(14) = 1 - (X(2) * X(3)) / 40

 

217 X(15) = 1 - (5 * X(2)) / X(1)

 

218 X(16) = 1 - (X(1)) / (12 * X(2))

 

219 X(17) = 1 - (1.5 * X(6) + 1.9) / X(4)

 

220 X(18) = 1 - (1.1 * X(7) + 1.9) / X(5)

 

221 IF X(1) < 2.6 THEN 1670

222 IF X(1) > 3.6 THEN 1670
223 IF X(2) < .7 THEN 1670

224 IF X(2) > .8 THEN 1670
225 IF X(3) < 17 THEN 1670

226 IF X(3) > 28 THEN 1670

235 IF X(4) < 7.3 THEN 1670

236 IF X(4) > 8.3 THEN 1670
245 IF X(5) < 7.8 THEN 1670

246 IF X(5) > 8.3 THEN 1670

247 IF X(6) < 2.9 THEN 1670

248 IF X(6) > 3.9 THEN 1670
249 IF X(7) < 5 THEN 1670

250 IF X(7) > 5.5 THEN 1670

 

268 FOR J99 = 8 TO 18

 

269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

270 NEXT J99

 

333 POBA = -.7854 * X(1) * X(2) ^ 2 * (3.3333 * X(3) ^ 2 + 14.9334 * X(3) - 43.0934) + 1.508 * X(1) * (X(6) ^ 2 + X(7) ^ 2) - 7.477 * (X(6) ^ 3 + X(7) ^ 3) - .7854 * (X(4) * X(6) ^ 2 + X(5) * X(7) ^ 2) + 1000000 * (X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) + X(16) + X(17) + X(18))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 18

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -999999 THEN 1999

 

1900 PRINT A(1), A(2), A(3)

1903 PRINT A(4), A(5), A(6)
1906 PRINT A(7), A(8), A(9)

1907 PRINT A(10), A(11), A(12)

1908 PRINT A(13), A(14), A(15)
1909 PRINT A(16), A(17), A(18)

1912 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [21]. The complete output through JJJJ = -31676.94000005171 is shown below:

3.499999791773702       .6999999582825123       17
7.299999952335762       7.799999952390007       3.35021466682406
5.286683229900787       0       0
0       0       0
0       0       0
0       0       0
-2996.218141343378       -31676.94000005171

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. One can compare this solution at JJJJ= -31676.94000005171 with the solutions in Table 6 of Ong Kok Meng et al. [12, p. 10]. One notes line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [21], the wall-clock time for obtaining the output through JJJJ= -31676.94000005171 was four minutes, total.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .

[3] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.

[5] 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.

[6] 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.

[7] 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.

[8] 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.

[9] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.9

[10] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[11] 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/

[12] Ong Kok Meng, Ong Pauline, Sia Chee Kiong, H. A. Wahab, N. Jafferi. Application of Modified Flower Pollination Algorithm on Mechanical Engineering Design Problem. IOP Conf. Series: Materials Science and Engineering 165 (2017) 012032.

[13] 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.

[14] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[15] 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.

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

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

[18] 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

[19] 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.

[20] 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.

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

[22] 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/

[23] 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.

 

Solving a Griewank Problem in 2500 Continuous Variables Using the Method of This Blog

Jsun Yui Wong

The computer program listed below works on the following Griewank problem based on a problem in Gandomi et al. [5, p. 93]. (Also see www.mathworld.wolfram.com/GriewankFunction/html.)

Minimize

2500 2500
1+(1 / 4000)*SIGMA X(i)^2-PI COS(X(i)/i^.5 )
i=1 i=1

where -600<=X(i)<=600, i=1, 2, 3,..., 2500.

 

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(2599), N(2599), A(2599), H(2599), L(2599), U(2599), X(2511), D(2511), P(2511), PS(2533)

12 FOR JJJJ = -32000 TO 32000 STEP .01

 

14 RANDOMIZE JJJJ
16 M = -1D+37

75 FOR J44 = 1 TO 2500

 

76 A(J44) = -600 + RND * 1200

 

79 NEXT J44

 

128 FOR I = 1 TO 30

 

129 FOR KKQQ = 1 TO 2500

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 2500)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

193 FOR J44 = 1 TO 2500

 

221 IF X(J44) < -600 THEN 1670

 

222 IF X(J44) > 600 THEN 1670

 

223 NEXT J44
241 SUMOF = 0
244 FOR J66 = 1 TO 2500

247 SUMOF = SUMOF + X(J66) ^ 2

249 NEXT J66

 

292 PRODOF = 1

293 FOR J77 = 1 TO 2500

295 PRODOF = PRODOF * COS(X(J77) / J77 ^ .5)

 

296 NEXT J77

 

329 POBA = -1 - (1 / 4000) * (SUMOF) + (PRODOF)

 

 

466 p = POBA

1111 IF p <= M THEN 1670

 

1452 M = p
1454 FOR KLX = 1 TO 2500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1499 REM PRINT A(1), A(1200), M, JJJJ

 

1557 GOTO 128

1670 NEXT I
1889 REM IF M < -.01 THEN 1999

 1900 PRINT A(1), A(2), A(2499)

1936 PRINT A(2500), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [20]. The complete output through JJJJ= -31999.99 is shown below:

-1.555497972635494D-09       -4.083494305531338D-08       6.6557157161668D-08
1.590678819399186D-07       -1.36273829677222D-11       -32000

-5.056513435954574D-09       6.59685482758314D-09       -1.91060451308703D-08
-1.18520770921616D-08       -3.718631305660303D-13       -31999.99

 Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. Only the A's of line 1900 and line 1936 are shown above. M=0 is optimal; see Gandomi, Yang, Talatahari, and Alavi [5, p. 93].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [20], the wall-clock time for obtaining the output through JJJJ=-31999.99 was 13 minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .

[3] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.

[5] Amir Hossein Gandomi, Xin-She Yang, Siamak Talatahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.

[6] 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.

[7] 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.

[8] 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.

[9] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.9

[10] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[11] 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/

[12] 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.

[13] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[14] 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.

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

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

[17] 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

[18] 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.

[19] 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.

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

[21] 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/

[22] 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.

Solving a Himmelblau Benchmark Problem Using the Method of This Blog, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the Himmelblau problem on pp. 19-20 of Gandomi, Yang, and Alavi [6]; one takes note of the typos [7].

Whereas line 128 of the earlier edition is 128 FOR I = 1 TO 5000, here line 128 is 128 FOR I = 1 TO 10000.

 

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 FOR J44 = 1 TO 5

 

74 A(J44) = 27 + RND * 75

 

79 NEXT J44

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 5

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 5)

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

200 X(6) = 92 - 85.334407 - .0056858 * X(2) * X(5) - .0006262 * X(1) * X(4) + .0022053 * X(3) * X(5)

 

202 X(7) = 110 - 80.51249 - .0071317 * X(2) * X(5) - .0029955 * X(1) * X(2) - .0021813 * X(3) * X(3)

 

 

204 X(8) = 25 - 9.300961 - .0047026 * X(3) * X(5) - .0012547 * X(1) * X(3) - .0019085 * X(3) * X(4)

 

207 X(9) = 0 + 85.334407 + .0056858 * X(2) * X(5) + .0006262 * X(1) * X(4) - .0022053 * X(3) * X(5)

 

209 X(10) = -90 + 80.51249 + .0071317 * X(2) * X(5) + .0029955 * X(1) * X(2) + .0021813 * X(3) * X(3)

  

211 X(11) = -20 + 9.300961 + .0047026 * X(3) * X(5) + .0012547 * X(1) * X(3) + .0019085 * X(3) * X(4)

 

221 IF X(1) < 78 THEN 1670

222 IF X(1) > 102 THEN 1670
223 IF X(2) < 33 THEN 1670

224 IF X(2) > 45 THEN 1670
225 IF X(3) < 27 THEN 1670

226 IF X(3) > 45 THEN 1670

235 IF X(4) < 27 THEN 1670

236 IF X(4) > 45 THEN 1670
245 IF X(5) < 27 THEN 1670

246 IF X(5) > 45 THEN 1670

 

268 FOR J99 = 6 TO 11

 

269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

270 NEXT J99

 

330 POBA = 40792.141 - 5.3578547 * X(3) * X(3) - .8356891 * X(1) * X(5) - 37.293239 * X(1) + 1000000 * X(6) + 1000000 * X(7) + 1000000 * X(8) + 1000000 * X(9) + 1000000 * X(10) + 1000000 * X(11)

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 11

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < 30665.5 THEN 1999

 

1900 PRINT A(1), A(2), A(3)

1903 PRINT A(4), A(5), A(6)
1906 PRINT A(7), A(8), A(9)

1908 PRINT A(10), A(11)

 1909 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [20]. The complete output through JJJJ= -31997.79000000035 is shown below:

78.00000000000038       33.00005651385925       29.99530102021148
44.99999973147974       36.77569923614965        0
0       0       0
0       0
30665.53161902383       -31998.1200000003

78.00000000000007       33.00020179728288       29.99536801302564
44.99999938131447       36.7755299727477        0
0       0       0
0       0
30665.52111933047       -31997.79000000035

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

The candidate solution above at JJJJ= -31998.1200000003 is comparable to the solutions presented in Table 2 of Gandomi, Yang, and Alavi [6, p. 20].

One notes that with the present algorithm none of the six constraints is violated because of line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [20], the wall-clock time for obtaining the output through JJJJ= -31997.79000000035 was one minute, total.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .

[3] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.

[5] 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.

[6] 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.

[7] 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.

[8] 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.

[9] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.9

[10] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[11] 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/

[12] 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.

[13] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[14] 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.

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

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

[17] 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

[18] 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.

[19] 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.

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

[21] 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/

[22] 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.

 

Solving a Griewank Test Problem in 1500 Continuous Variables Using the Method of This Blog

 Jsun Yui Wong

The computer program listed below works on the following Griewank problem based on a problem in Gandomi et al. [5, p. 93]. (Also see www.mathworld.wolfram.com/GriewankFunction/html.)

Minimize

                         1500                  1500
1+(1 / 4000)*SIGMA X(i)^2-PI COS(X(i)/i^.5 )
                          i=1                     i=1

where -600<=X(i)<=600, i=1, 2, 3,..., 1500.

 

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(1399), N(1599), A(1599), H(1599), L(1599), U(1599), X(1511), D(1511), P(1511), PS(1533)

12 FOR JJJJ = -32000 TO 32000 STEP .01

14 RANDOMIZE JJJJ
16 M = -1D+37

75 FOR J44 = 1 TO 1500

 

76 A(J44) = -600 + RND * 1200

 

79 NEXT J44

 

128 FOR I = 1 TO 20

 

 

129 FOR KKQQ = 1 TO 1500

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 1500)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

193 FOR J44 = 1 TO 1500

 

221 IF X(J44) < -600 THEN 1670

 

222 IF X(J44) > 600 THEN 1670

 

223 NEXT J44
241 SUMOF = 0
244 FOR J66 = 1 TO 1500

247 SUMOF = SUMOF + X(J66) ^ 2

249 NEXT J66

 

292 PRODOF = 1

293 FOR J77 = 1 TO 1500

295 PRODOF = PRODOF * COS(X(J77) / J77 ^ .5)

 

296 NEXT J77

 

329 POBA = -1 - (1 / 4000) * (SUMOF) + (PRODOF)

 

 

466 p = POBA

1111 IF p <= M THEN 1670

 

1452 M = p
1454 FOR KLX = 1 TO 1500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1499 REM PRINT A(1), A(1200), M, JJJJ

 

1557 GOTO 128

1670 NEXT I
1889 REM IF M < -.002 THEN 1999

 

1900 PRINT A(1), A(2), A(1499)

1935 PRINT A(1500), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [20]. The complete output through JJJJ= -31999.96000000001 is shown below:

-2.0069475487046D-08       -3.820605138020894D-08       -1.51981001944121D-05
1.84230210131093D-06       -5.904050545714735D-02       -32000

-2.877798198316777D-07       8.342101904723945D-09       6.174370755255956D-09
-3.440040754669161D-08       -1.078134489237298D-11       -31999.99

-3.629654497750748D-08       -4.306689541724595D-08       5.840468962767134D-08
-2.022673780609257D-07       -6.741319742015195D-13       -31999.98

-1.268823132889174       3.059449433936223       -1.07180906483747
.3338238094773224       -3.614124936831467       -31999-97000000001

-8.97377169827229D-09       3.137996583265033D-10       4.189476350013155D-08
1.80806067723507D-08       -5.549233273415D-12       -31999.96000000001

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. Only the A's of line 1900 and line 1935 are shown above. The best candidate solution shown above is at JJJJ=-31999.98 with M= -6.741319742015195D-13. M=0 is optimal; see Gandomi, Yang, Talatahari, and Alavi [5, p. 93].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [20], the wall-clock time for obtaining the output through JJJJ=-31999.96000000001 was nine minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .

[3] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.

[5] 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.

[6] 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.

[7] 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.

[8] 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.

[9] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.9

[10] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[11] 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/

[12] 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.

[13] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[14] 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.

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

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

[17] 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

[18] 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.

[19] 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.

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

[21] 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/

[22] 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.

 

Solving a Sphere Problem in 1000 Continuous Variables Using the Method of This Blog

Jsun Yui Wong

The computer program listed below works on the following Sphere problem based on a problem in Gandomi et al. [5, p. 93].

Minimize

  1000
(SIGMA X(i)^2)^.5
   i=1

where -100<=X(i)<=100, i=1, 2, 3,..., 1000.

 

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(2599), N(2599), A(2599), H(2599), L(2599), U(2599), X(2511), D(2511), P(2511), PS(2533)

12 FOR JJJJ = -32000 TO 32000 STEP .01

 

14 RANDOMIZE JJJJ
16 M = -1D+37

75 FOR J44 = 1 TO 1000

 

76 A(J44) = -100 + RND * 200

 

79 NEXT J44

 

128 FOR I = 1 TO 15

 

129 FOR KKQQ = 1 TO 1000

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 1000)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

193 FOR J44 = 1 TO 1000

 

221 IF X(J44) < -100 THEN 1670

 

222 IF X(J44) > 100 THEN 1670

 

223 NEXT J44
241 SUMOF = 0
244 FOR J66 = 1 TO 1000

247 SUMOF = SUMOF + X(J66) ^ 2

249 NEXT J66
332 POBA = -(SUMOF) ^ .5

 

466 p = POBA

1111 IF p <= M THEN 1670

 

1452 M = p

 

1454 FOR KLX = 1 TO 1000

 

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1499 REM PRINT A(1), A(1200), M, JJJJ

 

1557 GOTO 128

1670 NEXT I
1889 IF M < -.000001 THEN 1999

1900 PRINT A(1), A(2), A(999)

1936 PRINT A(1000), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [20]. The complete output through JJJJ= -31999.80000000003 is shown below:

1.242473448353766D-10          -3.169130824389183D-11         -4.832616724076011D-11
-2.416823451193115D-10          -1.636619136753577D-07          -31999.91000000001

-1.44871059677738D-14          -2.826738616185D-14          4.420007916792124D-13
-5.688160280552673D-14          -5.106626572380846D-11          -31999.80000000003

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. Only the A's of line 1900 and line 1936 are shown above. M=0 is optimal; see Gandomi, Yang, Talatahari, and Alavi [5, p. 93].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [20], the wall-clock time for obtaining the output through JJJJ=-31999.80000000003 was seven minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .

[3] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.

[5] 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.

[6] 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.

[7] 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.

[8] 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.

[9] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.9

[10] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[11] 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/

[12] 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.

[13] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[14] 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.

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

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

[17] 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

[18] 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.

[19] 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.

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

[21] 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/

[22] 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.

 

Solving a Griewank Problem in 2500 General Integers by Using the Method of This Blog, Third Edition

Jsun Yui Wong

The computer program listed below works on the following Griewank problem based on a problem in Gandomi et al. [5, p. 93]. (Also see www.mathworld.wolfram.com/GriewankFunction/html.)

Minimize

                         n                    n
1+(1 / 4000)*SIGMA X(i)^2-PI COS(X(i)/i^.5 )
                         i=1                 i=1

where -600<=X(i)<=600 and each X(i) is an integer, i=1, 2, 3,..., 2500.

 

0 DEFDBL A-Z

2 DEFINT K, X

 

3 DIM B(2599), N(2599), A(2599), H(2599), L(2599), U(2599), X(2511), D(2511), P(2511), PS(2533)

12 FOR JJJJ = -32000 TO 32000 STEP .01

 

14 RANDOMIZE JJJJ
16 M = -1D+37

75 FOR J44 = 1 TO 2500

 

76 A(J44) = -600 + RND * 1200

 

79 NEXT J44

 

128 FOR I = 1 TO 20000

 

129 FOR KKQQ = 1 TO 2500

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 2500)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

193 FOR J44 = 1 TO 2500

 

221 IF X(J44) < -600 THEN 1670

 

222 IF X(J44) > 600 THEN 1670

 

223 NEXT J44
241 SUMOF = 0
244 FOR J66 = 1 TO 2500

247 SUMOF = SUMOF + X(J66) ^ 2

249 NEXT J66

 

292 PRODOF = 1

293 FOR J77 = 1 TO 2500

295 PRODOF = PRODOF * COS(X(J77) / J77 ^ .5)

 

296 NEXT J77

 

329 POBA = -1 - (1 / 4000) * (SUMOF) + (PRODOF)

 

 

466 p = POBA

1111 IF p <= M THEN 1670

 

1452 M = p
1454 FOR KLX = 1 TO 2500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1499 REM PRINT A(1), A(1200), M, JJJJ

 

1557 GOTO 128

1670 NEXT I
1889 IF M < -.01 THEN 1999

 

1900 PRINT A(1), A(2), A(3)

 

1933 PRINT A(94), A(95), A(96)

1935 PRINT A(1197), A(1198), A(2499)

 

1936 PRINT A(2500), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [20]. The complete output through JJJJ= -31999.88000000002 is shown below:

0       0       0
0       0       0
0       0       0
0       0       -32000

0       0       0
0       0       0
0       0       0
0       -5.894241097741668D-04       -31999.98

0 0 0
0 0 0
0 0 0
0 -5.474272538122103D-04 -31999.96000000001

0 0 0
0 0 0
0 0 0
0 0 -31999.95000000001

0 0 0
0 0 0
0 0 0
0 -5.474272538122103D-04 -31999.94000000001

0 0 0
0 0 0
0 0 0
0 0 -31999.93000000001

0 0 0
0 0 0
0 0 0
0 0 -31999.90000000002

0       0       0
0       0       0
0       0       0
0       0       -31999.89000000002

0       0       0
0       0       0
0       0       0
0       -4.55724920516279D-03       -31999.88000000002

 

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. Only the A's of line 1900 through line 1936 are shown above. M=0 is optimal; see Gandomi, Yang, Talatahari, and Alavi [5, p. 93].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [20], the wall-clock time for obtaining the output through JJJJ=31999.88000000002 was 52 minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .

[3] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.

[5] Amir Hossein Gandomi, Xin-She Yang, Siamak Talatahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.

[6] 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.

[7] 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.

[8] 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.

[9] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.9

[10] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[11] 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/

[12] 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.

[13] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[14] 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.

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

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

[17] 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

[18] 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.

[19] 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.

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

[21] 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/

[22] 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.

 

Solving a Griewank Problem in 1500 General Integers by Using the Method of This Blog, Second Edition

Jsun Yui Wong

The computer program listed below works on the following Griewank problem based on a problem in Gandomi et al. [5, p. 93]. (Also see www.mathworld.wolfram.com/GriewankFunction/html.)

Minimize

                         n                    n
1+(1 / 4000)*SIGMA X(i)^2-PI COS(X(i)/i^.5 )
                          i=1                i=1

where -600<=X(i)<=600 and each X(i) is an integer, i=1, 2, 3,..., 1500.

 

0 DEFDBL A-Z

2 DEFINT K, X

 

3 DIM B(1399), N(1599), A(1599), H(1599), L(1599), U(1599), X(1511), D(1511), P(1511), PS(1533)

12 FOR JJJJ = -32000 TO 32000 STEP .01

 

14 RANDOMIZE JJJJ
16 M = -1D+37

75 FOR J44 = 1 TO 1500

 

76 A(J44) = -600 + RND * 1200

 

79 NEXT J44

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 1500

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 1500)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

193 FOR J44 = 1 TO 1500

 

221 IF X(J44) < -600 THEN 1670

 

222 IF X(J44) > 600 THEN 1670

 

223 NEXT J44
241 SUMOF = 0
244 FOR J66 = 1 TO 1500

247 SUMOF = SUMOF + X(J66) ^ 2

249 NEXT J66

 

292 PRODOF = 1

293 FOR J77 = 1 TO 1500

295 PRODOF = PRODOF * COS(X(J77) / J77 ^ .5)

 

296 NEXT J77

 

329 POBA = -1 - (1 / 4000) * (SUMOF) + (PRODOF)

 

 

466 p = POBA

1111 IF p <= M THEN 1670

 

1452 M = p
1454 FOR KLX = 1 TO 1500

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1499 REM PRINT A(1), A(1200), M, JJJJ

 

1557 GOTO 128

1670 NEXT I
1889 REM IF M < -.002 THEN 1999

 

1900 PRINT A(1), A(2), A(3)

 

1933 PRINT A(94), A(95), A(96)

1935 PRINT A(1197), A(1198), A(1499)

 

1936 PRINT A(1500), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [20]. The output through JJJJ=-31999.88000000002 is summarized below:

0       0       0
0       0       0
0       0       0
0       -6.604106476972102D-02       -32000
.
.
.

0       0       0
0       0       0
0       0       0
0       0       -31999.89000000002

0       0       0
0       0       0
0       0       0
0       0       -31999.88000000002

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. Only the A's of line 1900 through line 1936 are shown above.  Tthe candidate solution at JJJJ=-31999.89000000002 is the first occurrence of optimality of the computer run.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [20], the wall-clock time for obtaining the output through JJJJ= -31999.890000000002 was 20 minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .

[3] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.

[5] 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.

[6] 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.

[7] 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.

[8] 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.

[9] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.9

[10] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[11] 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/

[12] 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.

[13] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[14] 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.

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

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

[17] 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

[18] 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.

[19] 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.

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

[21] 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/

[22] 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.

 

Optimizing a Griewank Problem Having n=1200 by Using the Method of This Blog, Third Edition

 

Jsun Yui Wong

The computer program listed below works on the following Griewank problem based on page 93 of Gandomi et al. [5]. (Also see www.mathworld.wolfram.com/GriewankFunction/html.)

Minimize

                          n                           n
1+(1 / 4000)*SIGMA X(i)^2-PI COS(X(i)/i^.5 )
                          i=1                       i=1

-600<=X(i)<=600.  

The present case has n=1200.

 

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(1399), N(1399), A(1399), H(1399), L(1399), U(1399), X(1311), D(1311), P(1311), PS(1333)

12 FOR JJJJ = -32000 TO 32000 STEP .01

 

14 RANDOMIZE JJJJ
16 M = -1D+37

75 FOR J44 = 1 TO 1200

 

76 A(J44) = -600 + RND * 1200

 

79 NEXT J44

 

128 FOR I = 1 TO 20

 

129 FOR KKQQ = 1 TO 1200

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 1200)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

193 FOR J44 = 1 TO 1200

 

221 IF X(J44) < -600 THEN 1670

 

222 IF X(J44) > 600 THEN 1670

 

223 NEXT J44
241 SUMOF = 0
244 FOR J66 = 1 TO 1200

247 SUMOF = SUMOF + X(J66) ^ 2

249 NEXT J66

 

292 PRODOF = 1

293 FOR J77 = 1 TO 1200

295 PRODOF = PRODOF * COS(X(J77) / J77 ^ .5)

 

296 NEXT J77

 

329 POBA = -1 - (1 / 4000) * (SUMOF) + (PRODOF)

 

 

466 p = POBA

1111 IF p <= M THEN 1670

 

1452 M = p
1454 FOR KLX = 1 TO 1200

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -.002 THEN 1999

 

1900 PRINT A(1), A(2), A(3)

 

1903 REM PRINT A(4), A(5), A(6)

 

1933 REM PRINT A(94), A(95), A(96)

1935 REM PRINT A(1197), A(1198), A(1199)

 

1936 PRINT A(1200), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [20]. The complete output through JJJJ=-31999.98 is shown below:

-4.310795077151067D-09       -1.19782606221974D-09          6.343517393360775D-10                                                               
-2.529462756100607D-08       -5.003973364142711D-12       -32000

4.163966296149732D-08         1.931203253322993D-08       1.596217056817423D-08
-2.311917629622851D-08        -1.321639520913964D-12       -31999.98

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. Only the A's of line 1900 and line 1936 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [20], the wall-clock time for obtaining the output through JJJJ= -31999.98 was four minutes and a half, total.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

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

[21] 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/

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