Friday, June 22, 2018

Here Is Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems

Jsun Yui Wong

The computer program listed below seeks to solve the following mixed integer optimization problem from Deep et al.[11, pp. 514-515, Problem 14]:

Minimize       

9
sigma       (EXP(-(U(i) - X(2)) ^ X(3) / X(1)) - .01 * J44) ^ 2
i=1

where    U(i) = 25 + (-50 * LOG(.01 * i)) ^ (2 / 3)

subject to

       .1<=  X(1) is an integer variable <= 100
     
        0 <= X(2) is an integer variable <= 25.6

       0  <= X(3) is a continus variable <= 5. 


0 DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20)
5 FOR J44 = 1 TO 9

    7 U(J44) = 25 + (-50 * LOG(.01 * J44)) ^ (2 / 3)

8 NEXT J44

81 FOR JJJJ = -32000 TO 32000

    89 RANDOMIZE JJJJ
    90 M = -3E+30
    106 REM FOR J44 = 1 TO 15


    107 A(1) = 1 + FIX(RND * 100)

    109 A(2) = FIX(RND * 26)


    124 A(3) = RND * 5


    128 FOR I = 1 TO 5000


        129 FOR KKQQ = 1 TO 3
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        151 FOR IPP = 1 TO FIX(1 + RND * 2)
            153 J = 1 + FIX(RND * 3)

            155 IF J < 3 THEN GOTO 164 ELSE GOTO 156



            156 r = (1 - RND * 2) * A(J)
            158 X(J) = A(J) + (RND ^ (RND * 10)) * r

            161 GOTO 169


            164 IF RND < .5 THEN X(J) = A(J) - FIX(1 + RND * 2) ELSE X(J) = A(J) + FIX(1 + RND * 2)

        169 NEXT IPP
        311 REM GOTO 439
        312 X(1) = INT(X(1))


        313 IF X(1) < 1 THEN 1670

        314 IF X(1) > 100 THEN 1670

        317 REM NEXT J44
        322 X(2) = INT(X(2))


        323 IF X(2) < 0 THEN 1670
        324 IF X(2) > 25 THEN 1670

        325 IF X(3) < 0 THEN 1670

        326 IF X(3) > 5 THEN 1670


        395 SUMOBJ = 0

        397 FOR J44 = 1 TO 9
            398 SUMOBJ = SUMOBJ + (EXP(-(U(J44) - X(2)) ^ X(3) / X(1)) - .01 * J44) ^ 2

        399 NEXT J44

        441 PDU = -SUMOBJ

        466 P = PDU
        1111 IF P <= M THEN 1670
        1452 M = P
        1454 FOR KLX = 1 TO 3

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX


        1557 GOTO 128
    1670 NEXT I
    1889 REM IF M < .94561335 THEN 1999

    1904 PRINT A(1), A(2), A(3), M, JJJJ
 
1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [35]. The best candidate solutions through JJJJ =-31990 are shown below:

50      25      1.499999999999978                   -1.139164622473853D-27
-31998

50      25      1.49999999999999                     -2.687800656750728D-28
-31996

50      25      1.499999999999998                   -9.346625288811861D-30
-31992

50      25      1.5                                            -8.681731077433087D-33
-31991

50      25      1.5                                            -1.946889478860138D-30
-31990

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 [35], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31990 was 4 seconds, not including the time for “Creating .EXE file” (13 seconds, total, including the time for “Creating .EXE file” ).   One can compare the computational results above with those in
Deep et al. [11, p. 515, Problem 14].


Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

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[35]  Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[36] Jsun Yui Wong (2009, July 18).  An Integer Programming Computer Program Applied to One-Dimensional Space Allocation.  Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[37] Jsun Yui Wong (2009, December 18).  A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations.  Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[38] Jsun Yui Wong (2011, July 23).  A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem.  Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[39] Jsun Yui Wong (2011 July 27).   A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[40] Jsun Yui Wong (2012, April 24).  The Domino Method of General Integer Nonlinear Programming  Applied to Problem 10 of Lawler and Bell.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[41] Jsun Yui Wong (2012, September 27).  A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities.  Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[42]  Jsun Yui Wong (2013 January 10).  The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization.  http://myblogsubstance.typepad.com/substance/2013/01/

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