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].
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 5000
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=-31994.24000000092 is shown below:
78.00000000005112 33.00023762105803 29.99539016364582
44.99999513798106 36.77547568248082 0
0 0 0
0 0
30665.51753847873 -31994.24000000092
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 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=-31994.24000000092 was one minute, 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/
[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.
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