Jsun Yui Wong
The problem here is Li and Sun's Problem 14.3 but with 32760 unknowns instead of their 100 unknowns; see Li and Sun [12, pp. 414-415]. Their problem is based on Walukiewicz/Schittkowski Test Problem 282 [17, Test Problem 282, page 106 and page 23]. Specifically, the test example here is to minimize
32760-1
(X(1)-1)^2 + ( X(32760)-1)^2 + 32760* SIGMA (32760-i)* ( X(i)^2-X(i+1) )^2
i=1
subject to
-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,..., 32760.
This problem can be solved with the following procedure:
(1) There Is a large penalty when X(1)^7+X(2)^7+X(3)*7+...+X(32760)^7 is <32750 or when X(1)^7+X(2)^7+X(3)*7+...+X(32760)^7 is > 32770; see line 492 below.
The following computer program uses QB64 [19, 21].
0 DEFINT J, K, B, X, A
2 DIM A(32763), X(32763)
81 FOR JJJJ = -32000 TO 32000
85 LB = -FIX(RND * 6)
86 UB = FIX(RND * 6)
89 RANDOMIZE JJJJ
90 M = -1.5D+38
111 FOR J44 = 1 TO 32760
114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 32760
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)
167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32760
171 IF X(J44) < LB THEN X(J44) = LB
172 IF X(J44) > UB THEN X(J44) = UB
173 NEXT J44
221 SFE = 0
225 FOR J44 = 1 TO 32760
228 SFE = SFE + X(J44)^7
233 NEXT J44
251 TSL = -32750 + SFE
283 TSm = 32770 - SFe
297 IF TSL < 0 THEN TSL = TSL ELSE TSL = 0
367 IF TSm < 0 THEN TSm = TSm ELSE TSm = 0
400 SUMNEWZ = 0
403 FOR J44 = 1 TO 32759
405 SUMNEWZ = SUMNEWZ + (32760 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2
407 NEXT J44
411 SONE = -(X(1) - 1) ^ 2 - (X(32760) - 1) ^ 2 - 32760 * SUMNEWZ
492 PD1 = SONE + 5000000000# * TSL +5000000000*tsm
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32760
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1555 rem
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(3), A(32509), A(32760), M, JJJJ
1788 PRINT A(1111), A(11111), A(32757), A(32758), A(32759)
1999 NEXT JJJJ
Modelled after the computer programs in Wong [25], this BASIC computer program was run with QB64 [19, 21]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31992 is shown below:
-1 3 -3 3 3
-8.114774E+14 -32000
0 3 -1 -1 3
1 1 1 1 1
-4.280749E+08 -31999
1 1 1 1 1
-3 1 -3 -1 2
-8.890357E+14 -31998
0 -3 3 3 -1
0 0 0 0 0
-1.637553E+14 -31997
0 0 0 0 0
1 1 1 1 1
-1.910301E+09 -31996
1 1 1 1 1
2 -3 1 -3 -3
-1.031564E+15 -31995
1 2 -3 0 -1
-3 -3 -3 1 -3
-1.054913E+15 -31994
0 1 -3 -3 -3
3 1 3 -1 -1
-8.16768E+14 -31993
-2 3 -1 3 3
1 1 1 1 1
0 -31992
1 1 1 1 1
Above there is no rounding by hand; it is just straight copying by hand from the screen.
At JJJJ=-31992, M=0 is optimal. See Li and Sun [12, pp. 414-415].
Of the 32760 A's, only the ten A's of line 1778 and line 1788 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and with QB64 [19, 21], the wall-clock time for obtaining the output through JJJJ=-31992 was 24 hours.
(2) There Is a large penalty when X(1)^7+X(2)^7+X(3)*7+...+X(32760)^7 is between 32750 and 32770; see line 495 below.
The following computer program uses QB64 [19, 21].
0 DEFINT J, K, B, X, A
2 DIM A(32763), X(32763)
81 FOR JJJJ = -32000 TO 32000
85 LB = -FIX(RND * 6)
86 UB = FIX(RND * 6)
89 RANDOMIZE JJJJ
90 M = -1.5D+38
111 FOR J44 = 1 TO 32760
114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 32760
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32763)
167 IF RND < .5 THEN X(B) = (A(B) - 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32760
171 IF X(J44) < LB THEN X(J44) = LB
172 IF X(J44) > UB THEN X(J44) = UB
173 NEXT J44
221 SFE = 0
225 FOR J44 = 1 TO 32760
228 SFE = SFE + X(J44)^7
233 NEXT J44
251 TSL = -32750 + SFE
283 TSM = 32770 - SFE
297 REM IF TSL < 0 THEN TSL = TSL ELSE TSL = 0
367 REM IF TSm < 0 THEN TSm = TSm ELSE TSm = 0
388 IF TSL >0 AND TSM <0 THEN TSZ= -1000000 ELSE TSZ=0
400 SUMNEWZ = 0
403 FOR J44 = 1 TO 32759
405 SUMNEWZ = SUMNEWZ + (32760 - J44) * (X(J44) ^ 2 - X(J44 + 1)) ^ 2
407 NEXT J44
411 SONE = -(X(1) - 1) ^ 2 - (X(32760) - 1) ^ 2 - 32760 * SUMNEWZ
492 REM PD1 = SONE + 5000000000# * TSL +5000000000*tsm
495 PD1 = SONE + 5000000000# * TSZ
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32760
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1555 rem
1559 GOTO 128
1670 NEXT I
1778 PRINT A(1), A(2), A(3), A(32509), A(32760), M, JJJJ
1788 PRINT A(1111), A(11111), A(32757), A(32758), A(32759)
1999 NEXT JJJJ
Modelled after the computer programs in Wong [25], this BASIC computer program was run with QB64 [19, 21]. Copied by hand from the screen, the computer program's complete output through JJJJ=-31992 is shown below:
-1 1 1 1 3
-4.975006E+12 -32000
0 1 -1 1 2
0 0 0 2 5
-5.002075E+15 -31999
1 1 2 1 1
1 1 1 1 0
-5.002891E+15 -31998
0 1 2 0 0
1 1 1 1 1
-5.002059E+15 -31997
1 1 1 0 0
0 0 -1 -1 4
-4.919169E+12 -31996
1 1 -1 2 2
-1 1 0 0 -1
-4.348984E+12 -31995
1 0 0 -1 0
0 -1 1 -1 -1
-5.004152E+15 -31994
0 0 1 4 0
-1 1 1 1 -1
-5.165049E+12 -31993
1 1 1 1 0
0 0 0 0 0
-2 -31992
0 0 0 0 0
Above there is no rounding by hand; it is just straight copying by hand from the screen.
One notes the solution at JJJJ=-31992 with M=-2 because sometimes a nonoptimal solution is noteworthy; see Wah and Chen [20].
Of the 32760 A's, only the ten A's of line 1778 and line 1788 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and with QB64 [19, 21], the wall-clock time for obtaining the output through JJJJ=-31992 was 20 hours.
(3) The realized solution with M=0 of the first computer program at JJJJ=-31996--which was produced with the two artificial constraints X(1)^7 + X(2)^7 + X(3)^7
+ ... + X(32760)^7 >= 32750 and X(1)^7 + X(2)^7 + X(3)^7 + ... + X(32760)^7 <= 32770--is the best produced; hence it is adopted.
For a computer program involving a mix of continuous variables and integer variables, see Wong [23], for instance.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[19] E.K. Virtanen (2008-05-26). "Interview With Galleon",
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview
[20] Benjamin W. Wah, Yixin Chen. Solving Large-Scale Nonlinear Programming Problems by Constraint Partitioning. http://www.cse.wustl.edu/~ychen/public/C154.pdf
[21] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[22] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/
[23] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/
[24] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Elemsdition. http://myblogsubstance.typepad.com/substance/2013/09/
[25] Jsun Yui Wong (2014, June 27). A Unified Computer Program for Schittkowski's Test Problem 377, Second Edition. http://nonlinearintegerprogrammingsolver.blogspot.ca/2014_06_01_archive.html
[26] Jsun Yui Wong (2015, March 08). Nonconvex Mixed Integer Nonlinear Programming (MINLP) Computer Programs with a Divide-and-Conquer Strategy To Solve Li and Sun's Problem 14.3 but with n=15110 General Integer Variables. http://myblogsubstance.typepad.com/substance/2015/03/
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