Template for Solving Nonlinear/Linear Programming Problems Involving Absolute Values

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following problem:

Minimize ABS(X(1)) + 3 * ABS(X(2))

subject to

X(1) + 2 * X(2) >= 10,

X(1) - X(2) = 5.

The problem above is based on the numerical example in Granger, Yu, and Zhou [35].

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 2

112 A(J44) = -5 + FIX(RND * 11)

113 NEXT J44

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 * 1)

140 B = 1 + FIX(RND * 2)

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

192 X(1) = 5 + X(2)

194 IF X(1) + 2 * X(2) < 10 THEN 1670

342 PD1 = -ABS(X(1)) - 3 * ABS(X(2))

466 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

1779 IF M < -17 THEN 1999

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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through -31992 is shown below:

6.666666666666667 1.666666666666667 -11.66666666666667

-32000

6.666666666666667 1.666666666666667 -11.66666666666667

-31997

6.666666666666667 1.666666666666667 -11.66666666666667

-31992

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 = -31992 was 2 seconds, counting from "Starting program...".

**Acknowledgement**

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

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