Jsun Yui Wong
The computer program listed below seeks to solve the following Sjirk Boon [2] nonlinear system of equations used in Tsai and Lin [9]:
X(1) ^2+ X(3) ^ 2-1=0
X(2) ^2+ X(4) ^ 2 -1=0
X(5)*X(3) ^ 3 + X(6) * X(4) ^ 3 -1.2 =0
X(5) * X(1) ^ 3 + X(6) * X(2) ^ 3 - 1.2=0
X(5) * X(3) ^ 2 * X(1) + X(6) * X(4) ^ 2 * X(2) - .7=0
X(5) * X(3) * X(1) ^ 2 + X(6) * X(4) * X(2) ^ 2 - .7=0
-10<= X(i)<=10, i=1 to 6.
0 REM DEFDBL A-Z
2 DEFINT I, J, 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
14 RANDOMIZE JJJJ
16 M = -1D+37
64 FOR J44 = 1 TO 6
65 A(J44) = -1 + RND * 2
66 NEXT J44
126 REM IMAR=10+FIX(RND*32000)
128 FOR I = 1 TO 30000
129 FOR KKQQ = 1 TO 6
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))
181 J = 1 + FIX(RND * 6)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP
366 IF X(3) ^ 2 > 1 THEN GOTO 373
367 IF RND < .5 THEN X(1) = -(1 - X(3) ^ 2) ^ .5 ELSE X(1) = (1 - X(3) ^ 2) ^ .5
373 IF X(4) ^ 2 > 1 THEN GOTO 379
374 IF RND < .5 THEN X(2) = -(1 - X(4) ^ 2) ^ .5 ELSE X(2) = (1 - X(4) ^ 2) ^ .5
379 X(5) = (1.2 - X(6) * X(4) ^ 3) / X(3) ^ 3
386 PS(4) = X(5) * X(1) ^ 3 + X(6) * X(2) ^ 3 - 1.2
387 PS(5) = X(5) * X(3) ^ 2 * X(1) + X(6) * X(4) ^ 2 * X(2) - .7
388 PS(6) = X(5) * X(3) * X(1) ^ 2 + X(6) * X(4) * X(2) ^ 2 - .7
393 FOR J59 = 1 TO 6
394 IF X(J59) < -10 THEN GOTO 1670
395 IF X(J59) > 10 THEN GOTO 1670
398 NEXT J59
417 POBA = -ABS(PS(4)) - ABS(PS(5)) - ABS(PS(6))
459 POB1 = POBA
463 P1NEWMAY = POB1
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1453 PPOBA2 = POBA2
1454 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 REM GOTO 128
1670 NEXT I
1889 IF M < -.0003 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4)
1904 PRINT A(5), A(6), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [10]. The complete output through JJJJ=32111 is shown below:
-.915316 .4572198 -.4027364 1.040711
-1.442559 .9810117 -2.02933E-04 -31423
.4021955 -.9152884 .9155538 -.402799
1.440758 -1.44273 -2.408289E-04 -21010
-.5961376 .9154932 -1.355668 .4023333
-.4439624 1.441352 -8.565298E-05 -14235
.5550773 -.9154617 1.262391 -.4024051
.5497942 -1.441529 -4.222878E-05 4313
.8124468 .9154342 1.848123 .4024675
.1752132 1.441746 -1.225971E-05 9206
.4026785 -.9155846 .9153415 -.4021255
1.442545 -1.440738 -2.219319E-04 10583
-.4021617 -.9152779 -.91555686 -.4028229
-1.440658 -1.442819 -2.638689E-04 10628
-.4022661 .9153302 -.9155228 .4027041
-1.441015 1.442447 -1.750053E-04 12645
.5349895 -3.031289 1.217944 -1.333414
.6120817 -.0397175 -1.921458 E-04 18546
.4617415 .9153111 1.0513 .4027475
.9516512 1.442687 -2.454814E-04 23106
-.4872388 .9154198 -1.108508 .4025003
-.8119524 1.441869 -4.427775E-05 23690
.5734121 -.9154607 1.304184 -.4024074
.4986132 -1.441561 -3.151143E-05 31575
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 [10], the wall-clock time for obtaining the output through JJJJ=32111 was 90 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] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.
[5] 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.
[6] 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.
[7] 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.
[8] 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.
[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] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[11] 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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