Saturday, August 20, 2022

Four Similar Computer Programs for Solving Geometric Programming Problems, Including Those with Free Variables

 


Four Similar Computer Programs for Solving  Geometric Programming Problems, Including Those with Free Variables    


Jsun Yui Wong


The four computer programs listed below seek to solve all the geometric programming examples in Li and Tsai [51].


Example 1 in Li and Tsai, 2005 [51]


Similar to the computer program of the preceding paper, the first computer program listed below seeks to solve the formulation on page 9 of Li and Tsai [51] :


Minimize

      X(1) ^ 2.1 * X(2) * X(3) ^ 3 + X(1)

 subject to 

         -X(1) - X(2) ^ 2 <= -5 

         X(2) - X(1) + X(3) <= 13 

        0 <= X(1) <=3

         -2 <= X(2) <= 3 

         -2 <= X(3) <= 3.    

       


0 REM    DEFDBL A-Z


1 REM    DEFINT I, J, K

2 DIM B(99), N(99), A(200), H(99), L(99), U(99), X(200), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

9 FOR JJJJ = -32000 TO 32000


    89 RANDOMIZE JJJJ

    90 M = -3D+30



    111 A(1) = 0 + (RND * 3)



    112 A(2) = -2 + (RND * 5)


    113 A(3) = -2 + (RND * 5)







    128 FOR I = 1 TO 50000







        129 FOR KKQQ = 1 TO 3

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        139 FOR IPP = 1 TO FIX(1 + RND * 2)




            140 B = 1 + FIX(RND * 3)


            144 IF RND < .5 THEN 160 ELSE GOTO 167


            160 r = (1 - RND * 2) * A(B)


            164 X(B) = A(B) + (RND ^ (RND * 15)) * r

            165 GOTO 168


            167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 4) ELSE X(B) = A(B) + FIX(RND * 4)





        168 NEXT IPP

        196 IF X(1) < 0 THEN 1670

        197 IF X(1) > 3 THEN 1670



        198 IF X(2) < -2 THEN 1670

        199 IF X(2) > 3 THEN 1670


        200 IF X(3) < -2 THEN 1670

        201 IF X(3) > 3 THEN 1670


        219 IF -X(1) - X(2) ^ 2 > -5 THEN 1670


        221 IF X(2) - X(1) + X(3) > 13 THEN 1670



        459 PD1 = -X(1) ^ 2.1 * X(2) * X(3) ^ 3 - X(1)


        466 P = PD1


        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 1 TO 3

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX


    1670 NEXT I

    1777 IF M < 333 THEN 1999






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

1999 NEXT JJJJ



This computer program was run with qb64v1000-win [101].  Its complete output of one run through JJJJ= -31981 is shown below:



3      -2      3      539.4359      -31998


3      -2      3      539.4359      -31997


3      -2      3      539.4359      -31981


Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel Pentium CPU G620@ 2.60GHz, 4.00 GB of RAM (3.89 GB usable), 64-bit Operating System, and QB64v1000-win [101], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31981 was 3 seconds, counting from "Starting program...".  One can compare the computational results above with those in Li and Tsai [51].  



Example 2 in Li and Tsai, 2005 [51]


Similar to the computer program above, the computer program listed immediately below seeks to solve the formulation on page 11 of Li and Tsai [51] :


Minimize

       400 * X(1) ^ .9 + 1000 + 22 * (X(2) - 14.7) ^ 1.2 + X(4)


 subject to 

        

         X(2) = EXP(-3950 / (X(3) + 460) + 11.86)

        144 * (80 - X(3)) =  X(1) * X(4)  

          0 <= X(1) <= 15.1

         14.7 <= X(2) <= 94.2 

         -459.67 <= X(3) <= 80     

          0 <= X(4).


0 DEFDBL A-Z

1 REM    DEFINT I, J, K

2 DIM B(99), N(99), A(200), H(99), L(99), U(99), X(200), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

9 FOR JJJJ = -32000 TO 32000



    89 RANDOMIZE JJJJ

    90 M = -3D+30



    111 A(1) = 0 + (RND * 15.1)



    112 A(2) = 14.7 + (RND * 79.5)



    113 A(3) = -459.67 + (RND * 539.67)



    114 A(4) = (RND * 10)


    128 FOR I = 1 TO 50000


        129 FOR KKQQ = 1 TO 4

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        139 FOR IPP = 1 TO FIX(1 + RND * 3)


            140 B = 1 + FIX(RND * 4)


            144 IF RND < .5 THEN 160 ELSE GOTO 167


            160 r = (1 - RND * 2) * A(B)


            164 X(B) = A(B) + (RND ^ (RND * 15)) * r

            165 GOTO 168


            167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 4) ELSE X(B) = A(B) + FIX(RND * 4)


        168 NEXT IPP


        196 IF X(1) < 0 THEN 1670

        197 IF X(1) > 15.1 THEN 1670



        198 IF X(2) < 14.7 THEN 1670

        199 IF X(2) > 94.2 THEN 1670


        200 IF X(3) < -459.67 THEN 1670

        201 IF X(3) > 80 THEN 1670


        202 IF X(4) < 0 THEN 1670

      


        218 X(3) = (144 * 80 - X(1) * X(4)) / 144

        220 X(2) = EXP(-3950 / (X(3) + 460) + 11.86)


        459 PD1 = -400 * X(1) ^ .9 - 1000 - 22 * (X(2) - 14.7) ^ 1.2 - X(4)



        466 P = PD1


        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 1 TO 4

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX


    1670 NEXT I

    1777 IF M <= -9999999999 THEN 1999






    1904 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ


This computer program was run with qb64v1000-win [101].  Its complete output of one run through JJJJ= -31999 is shown below:


7.103713991400354D-18         94.1778659402041      80

8.136067055234835D-13        -5194.866244203784     -32000


2.404310389455815D-17         94.1778659402041      80

3.586516987154525D-13        -5194.866244203784     -31999


Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel Pentium CPU G620@ 2.60GHz, 4.00 GB of RAM (3.89 GB usable), 64-bit Operating System, and QB64v1000-win [101], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999 was 2 seconds, counting from "Starting program...".  One can compare the computational results above with those in Li and Tsai [51].  



Example 3 in Li and Tsai, 2005 [51]


The computer program listed immediately below seeks to solve the second formulation on page 11 of Li and Tsai [51] :


Minimize

       X(1) ^ -2 * X(2) ^ -.5 * X(3) ^ -1 + 8 * X(1) ^ -1 * X(4) ^ 2 - 8 * X(4)   


 subject to 

        X(1) - X(2) ^ .5 * X(3) ^ .5 <= 3 


         2 * X(1) + X(2) - X(3) + X(4) <= 6 

        

          1 <= X(1) <= 5

         3 <= X(2) <= 7 

         1 <= X(3) <= 10     

          1 <= X(4) <= 5. 


0 REM    DEFDBL A-Z


1 REM    DEFINT I, J, K

2 DIM B(99), N(99), A(200), H(99), L(99), U(99), X(200), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

9 FOR JJJJ = -32000 TO 32000






    89 RANDOMIZE JJJJ

    90 M = -3D+30



    111 A(1) = 1 + (RND * 4)



    112 A(2) = 3 + (RND * 4)


    113 A(3) = 1 + (RND * 9)



    114 A(4) = 1 + (RND * 4)








    128 FOR I = 1 TO 50000







        129 FOR KKQQ = 1 TO 4

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        139 FOR IPP = 1 TO FIX(1 + RND * 3)





            140 B = 1 + FIX(RND * 4)


            144 IF RND < .5 THEN 160 ELSE GOTO 167


            160 r = (1 - RND * 2) * A(B)


            164 X(B) = A(B) + (RND ^ (RND * 15)) * r

            165 GOTO 168


            167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 4) ELSE X(B) = A(B) + FIX(RND * 4)





        168 NEXT IPP

        196 IF X(1) < 1 THEN 1670

        197 IF X(1) > 5 THEN 1670



        198 IF X(2) < 3 THEN 1670

        199 IF X(2) > 7 THEN 1670

        200 IF X(3) < 1 THEN 1670

        201 IF X(3) > 10 THEN 1670


        202 IF X(4) < 1 THEN 1670

        203 IF X(4) > 5 THEN 1670


        235 IF X(1) - X(2) ^ .5 * X(3) ^ .5 > 3 THEN 1670


        237 IF 2 * X(1) + X(2) - X(3) + X(4) > 6 THEN 1670



        461 PD1 = -X(1) ^ -2 * X(2) ^ -.5 * X(3) ^ -1 - 8 * X(1) ^ -1 * X(4) ^ 2 + 8 * X(4)

        466 P = PD1


        1111 IF P <= M THEN 1670

        1452 M = P


        1454 FOR KLX = 1 TO 4

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX


    1670 NEXT I

    1777 IF M < -999999999 THEN 1999






    1904 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ


This computer program was run with qb64v1000-win [101].  Its complete output of one run through JJJJ= -31998 is shown below:



5      3.497652      9.999683      2.500422      9.997861

-32000


5      3.498328      9.99999      2.500001      9.997862

-31999


5      3.498937      9.999455      2.500042      9.997862

-31998


Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel Pentium CPU G620@ 2.60GHz, 4.00 GB of RAM (3.89 GB usable), 64-bit Operating System, and QB64v1000-win [101], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 2 seconds, counting from "Starting program...".  One can compare the computational results above with those in Li and Tsai [51].  


Example 4 in Li and Tsai, 2005 [51]


The computer program listed immediately below seeks to solve the formulation on page 12 of Li and Tsai [51] :


Minimize

        - ( -(X(4) - 1) ^ 2 - (X(5) - 2) ^ 2 - (X(6) - 1) ^ 2 + LOG(X(7) + 1) - (X(1) - 1) ^ 2 - (X(2) - 2) ^ 2 - (X(3) - 3) ^ 2)   


 subject to 


         X(4) + X(5) + X(6) + X(1) + X(2) + X(3) <= 5

        X(6) ^ 2 + X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 <= 5.5

         X(4) + X(1) <= 1.2 

         X(5) + X(2) <= 1.8 

         X(6) + X(3) <= 2.5 

         X(7) + X(1) <= 1.2 

         X(5) ^ 2 + X(2) ^ 2 <= 1.64 

         X(6) ^ 2 + X(3) ^ 2 <= 4.25 

         X(5) ^ 2 + X(3) ^ 2 <= 4.64       

          0 <= X(1) <= 1.2

         0 <= X(2) <= 1.8 

         0 <= X(3) <= 2.5     

         X(i) element  {0, 1},  i= 4, 5, 6, 7.


0 REM DEFDBL A-Z

1 REM    DEFINT I, J, K

2 DIM B(99), N(99), A(200), H(99), L(99), U(99), X(200), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

9 FOR JJJJ = -32000 TO 32000



    89 RANDOMIZE JJJJ

    90 M = -3D+30



    111 A(1) = 0 + (RND * 1.2)



    112 A(2) = 0 + (RND * 1.8)


    113 A(3) = 0 + (RND * 2.5)



    116 FOR J44 = 4 TO 7

        117 A(J44) = INT(RND)




    118 NEXT J44



    128 FOR I = 1 TO 50000




        129 FOR KKQQ = 1 TO 7

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        139 FOR IPP = 1 TO FIX(1 + RND * 5)



            140 B = 1 + FIX(RND * 7)


            144 IF RND < .5 THEN 160 ELSE GOTO 167


            160 r = (1 - RND * 2) * A(B)


            164 X(B) = A(B) + (RND ^ (RND * 15)) * r

            165 GOTO 168


            167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 4) ELSE X(B) = A(B) + FIX(RND * 4)




        168 NEXT IPP



        196 IF X(1) < 0 THEN 1670

        197 IF X(1) > 1.2 THEN 1670



        198 IF X(2) < 0 THEN 1670

        199 IF X(2) > 1.8 THEN 1670


        200 IF X(3) < 0 THEN 1670

        201 IF X(3) > 2.5 THEN 1670






        214 FOR J44 = 4 TO 7


            215 X(J44) = INT(X(J44))


        216 NEXT J44



        311 IF X(4) + X(1) > 1.2 THEN 1670


        312 IF X(5) + X(2) > 1.8 THEN 1670

        313 IF X(6) + X(3) > 2.5 THEN 1670


        314 IF X(7) + X(1) > 1.2 THEN 1670

        315 IF X(5) ^ 2 + X(2) ^ 2 > 1.64 THEN 1670

        316 IF X(6) ^ 2 + X(3) ^ 2 > 4.25 THEN 1670

        317 IF X(5) ^ 2 + X(3) ^ 2 > 4.64 THEN 1670



        318 IF X(6) ^ 2 + X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 > 5.5 THEN 1670

        319 IF X(4) + X(5) + X(6) + X(1) + X(2) + X(3) > 5 THEN 1670




        458 IF (X(7) + 1) < .0001 THEN 1670



        461 PD1 = -(X(4) - 1) ^ 2 - (X(5) - 2) ^ 2 - (X(6) - 1) ^ 2 + LOG(X(7) + 1) - (X(1) - 1) ^ 2 - (X(2) - 2) ^ 2 - (X(3) - 3) ^ 2


        466 P = PD1


        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 1 TO 7

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX


    1670 NEXT I

    1777 IF M < -5 THEN 1999




    1904 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), M, JJJJ

1999 NEXT JJJJ



This computer program was run with qb64v1000-win [101].  Its complete output of one run through JJJJ= -31994 is shown below:


.1978585      .7999998      1.907878      1      1

0      1      -4.583015      -31998


.2      .8      1.907878      1      1

0      1      -4.579583      -31997


.1999998      .8      1.907878      1      1

0      1      -4.579583      -31994


Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel Pentium CPU G620@ 2.60GHz, 4.00 GB of RAM (3.89 GB usable), 64-bit Operating System, and QB64v1000-win [101], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31994 was 3 seconds, counting from "Starting program...".  One can compare the computational results above with those in Li and Tsai [51].  



Acknowledgement

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

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