A computer program to solve nonconvex mixed-integer nonlinear programming problems: an illustration from Lundell, Skjal, and Westerlund [58]

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below attempts to solve the following mixed-integer nonlinear programming (MINLP) problem from Lundell, Skjal, and Westerlund [58, p. 136]:

Minimize

(2 * X(1) - 4) ^ 2 + (X(2) - 13 / 2) ^ 2

subject to

X(1) * COS(X(2)) ^ 2 + X(2) * SIN(X(1)) ^ 2 - 3 / X(2) + X(1) / 2 - 5 / 2 <= 0

2<= X(1) <= 4

2 <= X(2) <= 8

X(1) epsilon R

X(2) epsilon Z.

0 REM DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), G(128), J44(222), 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), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

87 RANDOMIZE JJJJ

88 M = -3D+30

108 A(1) = 2 + RND * 2

111 A(2) = 2 + FIX(RND * 7)

128 FOR i = 1 TO 50000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

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

141 B = 1 + FIX(RND * 2)

142 IF RND < 1.5 THEN 160 ELSE GOTO 167

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

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

164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

172 X(2) = INT(X(2))

246 IF X(1) * COS(X(2)) ^ 2 + X(2) * SIN(X(1)) ^ 2 - 3 / X(2) + X(1) / 2 - 5 / 2 > 0 THEN 1670

433 IF X(1) < 2 THEN 1670

434 IF X(1) > 4 THEN 1670

437 IF X(2) < 2 THEN 1670

439 IF X(2) > 8 THEN 1670

974 PD1 = -(2 * X(1) - 4) ^ 2 - (X(2) - 13 / 2) ^ 2

979 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1778 IF M < -999999999 THEN 1999

1888 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31990 is shown below:

**2.534119 5 -3.391133 -32000**

2.534119 5 -3.391133 -31999

2.000422 2 -20.25 -31998

2.000283 2 -20.25 -31997

2.000102 2 -20.25 -31996

2.534119 5 -3.391133 -31995

2.000348 2 -20.25 -31994

2.534119 5 -3.391133 -31993

2.000318 2 -20.25 -31992

2.534119 5 -3.391133 -31991

2.534119 5 -3.391133 -31990

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [103], the wall-clock time **(not CPU time)** for obtaining the output through JJJJ = -31990 was 3 seconds, counting from "Starting program...". One can compare the computational results above with those in Lundell, Skjal, and Westerlund [58, p. 137, Table 1].

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

**Acknowledgement**

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

References

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Acknowledgement

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

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