Jsun Yui Wong
The computer program listed below seeks to solve the following cantilever beam design problem in Yang et al.[13]:
Minimize .0624 * (X(1) + X(2) + X(3) + X(4) + X(5))
subject to 61 / (X(1) ^ 3) + 37 / (X(2) ^ 3) + 19 / (X(3) ^ 3) + 7 / (X(4) ^ 3) + 1 / (X(5) ^ 3) -1 <=0
0.01<= X(i) <=100, i=1, 2, 3, 4, 5.
The X(6) below is a slack variable.
One notes line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.
0 REM 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
64 FOR J44 = 1 TO 5
65 A(J44) = 1 + RND * 10
67 REM
68 REM
69 NEXT J44
128 FOR I = 1 TO 30000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 J = 1 + FIX(RND * 5)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP
241 X(6) = 1 - 61 / (X(1) ^ 3) - 37 / (X(2) ^ 3) - 19 / (X(3) ^ 3) - 7 / (X(4) ^ 3) - 1 / (X(5) ^ 3)
255 FOR J77 = 1 TO 5
257 IF X(J77) < .01 THEN 1670
258 IF X(J77) > 100 THEN 1670
259 NEXT J77
268 FOR J99 = 6 TO 6
269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0
270 NEXT J99
318 POBA = -.0624 * (X(1) + X(2) + X(3) + X(4) + X(5)) + 1000000 * X(6)
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -1.339957 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4)
1901 PRINT A(5), A(6), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [12]. The complete output through JJJJ=-31967.99 is shown below:
6.017338 5.308079 4.493506 3.501398
2.15334 0 -1.339956 -31967.99
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 [12], the wall-clock time for obtaining the output through JJJJ=-31967.99 was 11 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] 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.
[4] Ming-Hua Lin, Jung-Fa Tsai, Ming-Hua Lin (2014). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2014) 46:7, pp. 863-879.
[5] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.
[6] 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.
[7] 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.
[8] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[9] 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
[10] 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.
[11] 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.
[12] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[13] 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
[14] 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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