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,..., 1200.
0 DEFDBL A-Z
2 DEFINT K, X
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 10000
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
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)
1903 REM PRINT A(4), A(5), A(6)
1933 REM PRINT A(94), A(95), A(96)
1935 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=-31999.90000000002 is shown below:
-3 0 0
0 0 0
0 -2.923018515638994D-02 -32000
0 0 0
0 0 0
0 -3.687477901129832D-03 -31999.99
0 0 0
0 0 0
0 -9.591360299851246D-04 -31999.98
0 0 0
0 0 0
0 -1.241634136718078D-02 -31999.97000000001
0 0 0
0 0 0
0 -5.909225555920152D-02 -31999.96000000001
0 0 0
0 0 0
0 -1.137968021479058D-03 -31999.95000000001
0 0 0
0 0 0
0 0 -31999.94000000001
0 0 0
0 0 0
0 0 -31999.93000000001
0 0 0
0 0 0
0 0 -31999.92000000001
0 0 0
0 0 0
0 0 -31999.91000000001
3 0 0
0 0 0
0 -2.456846362867342D-02 -31999.90000000002
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, line 1935, 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= -31999.90000000002 was 11 minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[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.
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