Jsun Yui Wong
The computer program listed below seeks to solve the following problem in Lin and Tsai [5, pp. 873-875]:
Minimize 0.25 * PIE ^ 2 * X(2) * X(1) ^ 2 * X(3) + .5 * PIE ^ 2 * X(2) * X(1) ^ 2
subject to
4 * X(2) ^ 2 + 1.46 * X(1) * X(2) - 2.46 * X(1) ^ 2 - .5 * ((PIE * S) / (FMAX)) * (X(1) ^ 3 * X(2)) + .5 * ((PIE * S) / (FMAX)) * X(1) ^ 4<=0
8 * ((FMAX / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3)) + 1.05 * X(1) * X(3) + 2.1 * X(1) - LMAX<=0
DMIN - X(1)<=0
X(2) - DMAX<=0
3.0 - X(1) ^ (-1) * X(2)<=0
8 * ((FP / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3)) - SIGMAPM<=0
SIGMAW - 8 * (((FMAX - FP) / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3))<=0
where PIE = 3.141592654, FMAX = 1000, LMAX = 14.0, DMIN = .2, S = 189000.0, DMAX = 3.0, FP = 300.0, SIGMAPM = 6.0, SIGMAW = 1.25, and G = 11.5D+06,
X(1) equals .207, .225, .244, .263, .283, .307, .331, .362, .394, .4375, or .500,
0.6<= X(2) <=3,
1<= X(3) <=70 and integer.
X(4) through X(10) below are slack variables.
One notes line 269, which is 269 IF X(J99) < -.00001 THEN X(J99) = X(J99) ELSE X(J99) = 0; see Lin and Tsai [5, p. 875, Table 5].
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 32111 STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
21 PIE = 3.141592654: FMAX = 1000: LMAX = 14.0: DMIN = .2: S = 189000.0: DMAX = 3.0: FP = 300.0: SIGMAPM = 6.0: SIGMAW = 1.25: G = 11.5D+06
67 IF RND < .091 THEN A(1) = .207 ELSE IF RND < .1 THEN A(1) = .225 ELSE IF RND < .111 THEN A(1) = .244 ELSE IF RND < .125 THEN A(1) = .263 ELSE IF RND < .143 THEN A(1) = .283 ELSE IF RND < .167 THEN A(1) = .307 ELSE IF RND < .2 THEN A(1) = .331 ELSE IF RND < .25 THEN A(1) = .362 ELSE IF RND < .333 THEN A(1) = .394 ELSE IF RND < .5 THEN A(1) = .4375 ELSE A(1) = .500
68 A(2) = .6 + RND * 2.4
69 A(3) = 1 + FIX(RND * 70)
128 FOR I = 1 TO 10000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 J = 2 + FIX(RND * .2)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP
225 X(3) = INT(A(3))
239 X(4) = -4 * X(2) ^ 2 - 1.46 * X(1) * X(2) + 2.46 * X(1) ^ 2 + .5 * ((PIE * S) / (FMAX)) * (X(1) ^ 3 * X(2)) - .5 * ((PIE * S) / (FMAX)) * X(1) ^ 4
241 X(5) = -8 * ((FMAX / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3)) - 1.05 * X(1) * X(3) - 2.1 * X(1) + LMAX
245 X(6) = -DMIN + X(1)
249 X(7) = -X(2) + DMAX
251 X(8) = -3.0 + X(1) ^ (-1) * X(2)
253 X(9) = -8 * ((FP / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3)) + SIGMAPM
255 X(10) = -SIGMAW + 8 * (((FMAX - FP) / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3))
260 IF X(1) < .2 THEN GOTO 1670
261 IF X(2) < .6 THEN GOTO 1670
262 IF X(3) < 1 THEN GOTO 1670
263 REM IF X(4) < 10 THEN GOTO 1670
264 IF X(1) > .5 THEN GOTO 1670
265 IF X(2) > 3 THEN GOTO 1670
266 IF X(3) > 70 THEN GOTO 1670
267 REM IF X(4) > 200 THEN GOTO 1670
268 FOR J99 = 4 TO 10
269 IF X(J99) < -.00001 THEN X(J99) = X(J99) ELSE X(J99) = 0
270 NEXT J99
318 POBA = -.25 * PIE ^ 2 * X(2) * X(1) ^ 2 * X(3) - .5 * PIE ^ 2 * X(2) * X(1) ^ 2 + 1000000 * X(4) + 1000000 * X(5) + 1000000 * X(6) + 1000000 * X(7) + 1000000 * X(8) + 1000000 * X(9) + 1000000 * X(10)
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -2.99 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4)
1901 PRINT A(5), A(6), A(7), A(8)
1902 PRINT A(9), A(10), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [13]. The complete output through JJJJ=-31999.60000000006 is shown below:
.283 1.223037748512351 9 0
0 0 0 0
0 0 -2.658552077153712 -31999.60000000006
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The solution above is comparable to the 5 solutions presented in Table 5 of Lin and Tsai [5, p. 875].
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [13], the wall-clock time for obtaining the output through JJJJ=-31999.60000000006 was 10 seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[10] 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
[11] 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.
[12] 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.
[13] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[14] 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
[15] 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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