May 9, 2018 by 123writereadmath

Solving a Nonlinear Programming Problem Involving Discrete Variables and Continuous Variables

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Wang, Tsai, Ma, and Lee [52, Example 2]:

Minimize .6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(1) ^ 2 * X(4) + 19.84 * X(1) ^ 2 * X(3)

subject to

-X(1) + .0193 * X(3)<=0

-X(2) + .00954 * X(3)<=0

750 * 1728 -pie* X(3) ^ 2 * X(4) – (4 / 3) * pie * X(3) ^ 3<=0

-240<= X(4) <=0

1<=X(1) < = 1.375

.625<=X(2) < = 1.

48<=X(3) < = 52

90<=X(4) < = 112,

where X(1) and X(2) are discrete variables with discreteness of .625, and X(3) and X(4) are continuous variables.

X(5) through and X(7) below are slack variables added.

0 DEFDBL A-Z

2 DEFINT 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 32000 STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37

67 unitsize(1) = .0625
69 unitsize(2) = .0625
76 A(1) = 1 + FIX(RND * 7) * unitsize(1)

78 A(2) = .625 + FIX(RND * 7) * unitsize(2)

86 A(3) = 48 + (RND * 4)

88 A(4) = 90 + (RND * 22)

128 FOR I = 1 TO 20000

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

181 j = 1 + FIX(RND * 4)
182 IF j < 3 GOTO 187 ELSE GOTO 189

187 IF RND < .5 THEN X(j) = A(j) – FIX(RND * 10) * unitsize(j) ELSE X(j) = A(j) + FIX(RND * 10) * unitsize(j)
188 GOTO 222

189 r = (1 – RND * 2) * A(j)

190 X(j) = A(j) + (RND ^ (RND * 10)) * r

191 GOTO 222

222 NEXT IPP

274 IF X(1) < 1## THEN 1670

276 IF X(1) > 1.375## THEN 1670

284 IF X(2) < .625## THEN 1670

286 IF X(2) > 1## THEN 1670

294 IF X(3) < 48## THEN 1670

296 IF X(3) > 52## THEN 1670

297 IF X(4) < 90## THEN 1670

298 IF X(4) > 112## THEN 1670

321 X(5) = X(1) – .0193 * X(3)
322 X(6) = X(2) – .00954 * X(3)
326 X(7) = -750 * 1728 + 3.141592654 * X(3) ^ 2 * X(4) + (4 / 3) * 3.141592654 * X(3) ^ 3

401 FOR J47 = 5 TO 7

404 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

409 NEXT J47

451 POBA = -.6224 * X(1) * X(3) * X(4) – 1.7781 * X(2) * X(3) ^ 2 – 3.1661 * X(1) ^ 2 * X(4) – 19.84 * X(1) ^ 2 * X(3) + 1000000 * (X(5) + X(6) + X(7))

466 P = POBA

1111 IF P <= M THEN 1670

1450 M = P
1454 FOR KLX = 1 TO 7

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -7059.31 THEN 1999

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6)

1908 PRINT A(7), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [53]. The complete output through JJJJ = -31999.81000000003 is shown below:

1 .625 51.08013803893487 90.0001150938644
0 0
0 -7059.302881290298 -31999.95000000001

1 .625 51.08013700321705 90.0001228864803
0 0
0 -7059.302957554913 -31999.81000000003

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 [53], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31999.81000000003 was 5 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 of Wang, Tsai, Ma, and Lee [52, end of Section III].

Acknowledgment

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

References

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May 3, 2018 by 123writereadmath

Using Discrete Variables To Approximate Continuous Variables

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Tsai and Lin [51, p. 47, Example 3]:

Minimize X(1) * X(4) ^ 3 – X(3) – .5 * X(1) ^ 2 * X(2) ^ 4

subject to

X(1) * X(4) ^ 1.5 – X(2) – X(2) ^ .5 * X(3) ^ .4<=4

– X(1) – 2 * X(2)+ X(3)<=-2

0<= X(1) <=6

1<= X(2) <=10

1<= X(3) <=6

20<= X(4) <=30.

This “is a nonconvex problem with zero-value lower bound variables that can not be treated by current exponential-based methods,” Tsai and Lin (2008) [51, p. 47].

X(5) and X(6) below are slack variables added.

0 DEFDBL A-Z

2 DEFINT 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 32000 STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37

71 unitsize(1) = .001

74 unitsize(2) = .001
76 unitsize(3) = .001

79 unitsize(4) = .001

81 A(1) = 0 + FIX(RND * 1000) * unitsize(1)

83 A(2) = 1 + FIX(RND * 1000) * unitsize(1)
84 A(3) = 1 + FIX(RND * 1000) * unitsize(1)

85 A(4) = 20 + FIX(RND * 1000) * unitsize(1)

128 FOR I = 1 TO 20000

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 j = 1 + FIX(RND * 4)

187 IF RND < .5 THEN X(j) = A(j) – FIX(RND * 100) * unitsize(j) ELSE X(j) = A(j) + FIX(RND * 100) * unitsize(j)

222 NEXT IPP

268 IF X(1) < 0## THEN 1670

269 IF X(1) > 6## THEN 1670

274 IF X(2) < 1## THEN 1670

276 IF X(2) > 10## THEN 1670
278 IF X(3) < 1## THEN 1670

279 IF X(3) > 6## THEN 1670

288 IF X(4) < 20## THEN 1670

289 IF X(4) > 30## THEN 1670

323 X(6) = -2 + X(1) + 2 * X(2) – X(3)
327 X(5) = 4 – X(1) * X(4) ^ 1.5 + X(2) + X(2) ^ .5 * X(3) ^ .4
401 FOR J47 = 5 TO 6
405 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

409 NEXT J47

449 POBA = -X(1) * X(4) ^ 3 + X(3) + .5 * X(1) ^ 2 * X(2) ^ 4 + 1000000 * (X(5) + X(6))
466 P = POBA

1111 IF P <= M THEN 1670

1450 M = P
1454 FOR KLX = 1 TO 6

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -99 THEN 1999
1908 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [52]. The complete output through JJJJ = -31999.94000000001 shown below:

9.999999999998925D-04 10 6 20.001
0 0 -1.99620006000014 -32000

9.999999999999506D-04 9.999999999999995 5.999999999999996
20 0 0 -1.994999999999961
-31999.99

9.99999999999988D-04 9.999999999999996 6
20.001 0 0 -1.9962000600009
-31999.98

9.999999999999194D-04 9.999999999999996 5.999000000000001
20.001 0 0 -1.99720006000352
-31999.97000000001

9.999999999999914D-04 9.9990000000000002 5.999999999999997
20.001 0 0 -1.996202059700946
-31999.96000000001

1.734723475976807D-18 4.157 5.999999999999998
20.10900000000001 0 0 5.999999999999984
-31999.95000000001

4.336808689942018D-18 5.269000000000001 5.999999999999996
20.062 0 0 5.999999999999961
-31999.94000000001

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 [52], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =-31999.94000000001 was 2 seconds, not including the time for “Creating .EXE file” (10 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on p. 48 of Tsai and Lin (see also Table 1, p. 47, Example 3) [51].