Direct Way To Solve Signomial Geometric Programming Problems: Another Illustration

 

Direct Way To Solve Signomial Geometric Programming Problems: Another Illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following problem from Hou, Shen, and Chen [40, p. 11, Example 10]:   

Minimize            - ( -4 * (X(1) ^ 2 * X(3) + 2 * X(1) ^ 2 * X(2) * X(3) ^ 2 * X(5) + 2 * X(1) ^ 2 * X(2) * X(3)) * ((5 * X(1) ^ 2 * X(3) * X(4) ^ 2 * X(5) + 3 * X(2))) ^ (3 / 5) - 3 * (2 * X(4) ^ 2 * X(5) ^ 2) * (4 * X(1) ^ 2 * X(4) + 4 * X(2) * X(5)) ^ (5 / 3))    

subject to

      -2 * (2 * X(1) * X(5) + 5 * X(1) ^ 2 * X(2) * X(4) ^ 2 * X(5)) * ((3 * X(1) * X(4) * X(5) ^ 2 + 5 + 4 * X(3) * X(5) ^ 2)) ^ (1 / 2) <= -7684.470329, 

       2 * 2 * X(1) * X(2) ^ 2 * X(3) * X(4) ^ 2 * ((2 * X(1) * X(2) * X(3) * X(4) ^ 2 + 2 * X(2) * X(4) ^ 2 * X(5) - X(1) ^ 2 * X(5) ^ 2)) ^ (3 / 2) <= 1286590.314422 ,

       0<= X(1), X(2), X(3), X(4), X(5) <= 5.

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), 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)

85 FOR JJJJ = -32000 TO 32000

    89 RANDOMIZE JJJJ

    90 M = -3D+30

    111 FOR J44 = 1 TO 5

        112 A(J44) = FIX(RND * 6)

    113 NEXT J44

    128 FOR I = 1 TO 300000

        129 FOR KKQQ = 1 TO 5

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        139 FOR IPP = 1 TO FIX(1 + RND * 4)

            140 B = 1 + FIX(RND * 5)

            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 * 2) ELSE X(B) = A(B) + FIX(RND * 2)

        168 NEXT IPP

        171 IF X(1) < 0 THEN 1670

        172 IF X(2) < 0 THEN 1670

        173 IF X(3) < 0 THEN 1670

        174 IF X(4) < 0 THEN 1670

        175 IF X(5) < 0 THEN 1670

        177 IF X(1) > 5 THEN 1670

        178 IF X(2) > 5 THEN 1670

        179 IF X(3) > 5 THEN 1670

        181 IF X(4) > 5 THEN 1670

        182 IF X(5) > 5 THEN 1670

        344 IF -2 * (2 * X(1) * X(5) + 5 * X(1) ^ 2 * X(2) * X(4) ^ 2 * X(5)) * ((3 * X(1) * X(4) * X(5) ^ 2 + 5 + 4 * X(3) * X(5) ^ 2)) ^ (1 / 2) > -7684.470329 THEN 1670

        351 IF ((2 * X(1) * X(2) * X(3) * X(4) ^ 2 + 2 * X(2) * X(4) ^ 2 * X(5) - X(1) ^ 2 * X(5) ^ 2)) < 0## THEN 1670

        355 IF 2 * 2 * X(1) * X(2) ^ 2 * X(3) * X(4) ^ 2 * ((2 * X(1) * X(2) * X(3) * X(4) ^ 2 + 2 * X(2) * X(4) ^ 2 * X(5) - X(1) ^ 2 * X(5) ^ 2)) ^ (3 / 2) > 1286590.314422 THEN 1670

        392 PD1 = -4 * (X(1) ^ 2 * X(3) + 2 * X(1) ^ 2 * X(2) * X(3) ^ 2 * X(5) + 2 * X(1) ^ 2 * X(2) * X(3)) * ((5 * X(1) ^ 2 * X(3) * X(4) ^ 2 * X(5) + 3 * X(2))) ^ (3 / 5) - 3 * (2 * X(4) ^ 2 * X(5) ^ 2) * (4 * X(1) ^ 2 * X(4) + 4 * X(2) * X(5)) ^ (5 / 3)

        466 P = PD1

        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 1 TO 5

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

    1670 NEXT I

    1779 IF M < -28660 THEN 1999   

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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102].  Its complete output of one run through JJJJ= -31384 is summarized  below:

.........         .........        .........

.........         .........      -28627.53399853512

-31646

.........         .........        .........

.........         .........       -28623.92348786566

-31471

4.987229491583783         4.99981402284291             .1233794438347216

1.188778089642144         .9400432931887702          -28572.32267939633

-31384

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 [102], the wall-clock time (not CPU time) for obtaining the output through  JJJJ = -31384 was 12 minutes, counting from "Starting program...".   One can compare the computational results above with those in Hou, Shen, and Chen [40, p. 10, Table 2]. 

 

      

Acknowledgement

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

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Direct Way To Solve Signomial Geometric Programming Problems: An Illustration

 

Direct Way To Solve Signomial Geometric Programming Problems: An Illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following problem from Hou, Shen, and Chen [40, p. 9, Example 7]:   

Minimize          - 4 * X(2) + (X(1) - 1) ^ 2 + X(2) ^ 2 - 10 * X(3) ^ 2)

subject to

        X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 <= 2, 

        (X(1) - 2) ^ 2 + X(2) ^ 2 + X(3) ^ 2 <= 2, 

         2 - 2 ^ .5 <= X(1) <= 2 ^ .5,

         -2 ^ .5 <= X(2) <= 2 ^ .5,

         -2 ^ .5 <= X(3) <= 2 ^ .5.

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), 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)

85 FOR JJJJ = -32000 TO 32000

    89 RANDOMIZE JJJJ

    90 M = -3D+30

    111 FOR J44 = 1 TO 3

        112 A(J44) = FIX(RND * 11)

    113 NEXT J44

    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 * 10) ELSE X(B) = A(B) + FIX(RND * 10)

        168 NEXT IPP

        171 IF X(1) < 2 - 2 ^ .5 THEN 1670

        174 IF X(2) < -2 ^ .5 THEN 1670

        175 IF X(3) < -2 ^ .5 THEN 1670

        177 IF X(1) > 2 ^ .5 THEN 1670

        178 IF X(2) > 2 ^ .5 THEN 1670

        179 IF X(3) > 2 ^ .5 THEN 1670

        213 IF (X(1) - 2) ^ 2 + X(2) ^ 2 + X(3) ^ 2 > 2 THEN 1670

        219 IF (-X(1) ^ 2 - X(2) ^ 2 + 2) < 0## THEN 1670

        222 X(3) = (-X(1) ^ 2 - X(2) ^ 2 + 2) ^ .5

        271 IF X(1) < 2 - 2 ^ .5 THEN 1670

        274 IF X(2) < -2 ^ .5 THEN 1670

        275 IF X(3) < -2 ^ .5 THEN 1670

        277 IF X(1) > 2 ^ .5 THEN 1670

        278 IF X(2) > 2 ^ .5 THEN 1670

        279 IF X(3) > 2 ^ .5 THEN 1670

        280 IF (X(1) - 2) ^ 2 + X(2) ^ 2 + X(3) ^ 2 > 2 THEN 1670

        392 PD1 = 4 * X(2) - (X(1) - 1) ^ 2 - X(2) ^ 2 + 10 * X(3) ^ 2

        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

    1779 IF M < -999999999 THEN 1999

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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102].  Its complete output of one run through JJJJ= -31974 is shown below:

1      .1818181891346452      .9833321646828187

10.36363636363636         -31998

1      0       1        10     -31997

1      .1818181766443552      .9833321669922741

10.36363636363636         -31993

1      0       1        10     -31989

1      0       1        10     -31981

1      0       1        10     -31977

1.000000000000003      .1818181859018952      .9833321652805515

10.3636363636363         -31974

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 [102], the wall-clock time (not CPU time) for obtaining the output through  JJJJ = -31974 was 4 seconds, counting from "Starting program...".   One can compare the computational results above with those in Hou, Shen, and Chen [40, p. 10, Table 2]. 

     

Acknowledgement

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

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Straight Solving Geometric Programming Problems: An Illustration

 

Straight Solving Geometric Programming Problems: An Illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following problem from Wu [105, Illustrative Example]:   

Minimize          X(1) * X(2) ^ .5 * X(3) ^ 1.2 + 2 * X(1)

subject to

         2 * X(1) + X(2) + X(3) >= 8

         X(2) + 2 * X(3) >= 10.5

         X(1) + 2 * X(3) <= 10

     1<=  X(1), X(2), X(3) <=5

      

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), 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)

85 FOR JJJJ = -32000 TO 32000

    89 RANDOMIZE JJJJ

    90 M = -3D+30

    111 FOR J44 = 1 TO 3

        112 A(J44) = FIX(RND * 11)

    113 NEXT J44

    128 FOR I = 1 TO 150000

        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 * 10) ELSE X(B) = A(B) + FIX(RND * 10)

        168 NEXT IPP

        171 IF X(1) < 1## THEN 1670

        175 IF X(2) < 1## THEN 1670

        176 IF X(3) < 1## THEN 1670

        177 IF X(1) > 5## THEN 1670

        178 IF X(2) > 5## THEN 1670

        179 IF X(3) > 5## THEN 1670

        384 IF 2 * X(1) + X(2) + X(3) < 8## THEN 1670

        386 IF X(2) + 2 * X(3) < 10.5## THEN 1670

        388 IF X(1) + 2 * X(3) > 10## THEN 1670

        389 PD1 = -X(1) * X(2) ^ .5 * X(3) ^ 1.2 - 2 * 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

    1779 IF M < -9.45 THEN 1999

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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102].  Its complete output of one run through JJJJ= -31971 is shown below:

1         1.502220878415816      4.498889560792098

-9.448919749673387      -32000

1         1.500139847008161      4.499930076495974

-9.44582443893382      -31971

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 [102], the wall-clock time (not CPU time) for obtaining the output through  JJJJ = -31971 was 5 seconds, counting from "Starting program...".   

   

Acknowledgement

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

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Straight Solving Linear Fractional Programming Problems: An Illustration

 

Straight Solving Linear Fractional Programming Problems: An Illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following problem from Hasan and Acharjee [37, p. 12, Numerical Example 5]:   

Maximize      (X(1) + 2 * X(2) + 3.5 * X(3) + X(4) + 1) / (2 * X(1) + 2 * X(2) + 3.5 * X(3) + 3 * X(4) + 4)

  

subject to

         2 * X(1) + X(2) + 3 * X(3) + 3 * X(4) <= 10,

         X(1) + 2 * X(2) + X(3) + X(4) <= 14,

     

         X(1),  X(2), X(3),  X(4) >= 0.

  

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), 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)

85 FOR JJJJ = -32000 TO 32000

    89 RANDOMIZE JJJJ

    90 M = -3D+30

    111 FOR J44 = 1 TO 4

        112 A(J44) = FIX(RND * 21)

    113 NEXT J44

    128 FOR I = 1 TO 150000

        129 FOR KKQQ = 1 TO 4

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

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

            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 * 3) ELSE X(B) = A(B) + FIX(RND * 3)

        168 NEXT IPP

        171 IF X(1) < 0## THEN 1670

        175 IF X(2) < 0## THEN 1670

        177 IF X(3) < 0## THEN 1670

        179 IF X(4) < 0## THEN 1670

        404 IF 2 * X(1) + X(2) + 3 * X(3) + 3 * X(4) > 10## THEN 1670

        406 IF X(1) + 2 * X(2) + X(3) + X(4) > 14## THEN 1670

     

        409 PD1 = (X(1) + 2 * X(2) + 3.5 * X(3) + X(4) + 1) / (2 * X(1) + 2 * X(2) + 3.5 * X(3) + 3 * X(4) + 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

    1779 IF M < -9999 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 [102].  Its output of one run through JJJJ=-31899 is shown below:

0      6.40067550592987        1.198647516960092  

0      .85711984201531         -31982

1.391493333437622D-16      6.398962636323966       1.200345787777984  

2.065218353606715D-17      .857136976151587        -31906

0      6.399999127939391      1.200000290686772  

3.125690125099123D-19      .8571428521992004      -31899

One distinct solution is shown above.

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 [102], the wall-clock time (not CPU time) for obtaining the output through  JJJJ  = -31899 was 17 seconds, counting from "Starting program...".   

   

Acknowledgement

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

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