Jsun Yui Wong

The computer program listed below seeks to find one solution io the following multi-objective nonlinear programming problem:

Maximize -(-4.07 - 2.27 * X(1))

maximize -(-2.6 - .03 * X(1) - .02 * X(2) - (.01 / (1.39 - X(1) ^ 2)) - (.3 / (1.39 - X(2) ^ 2)))

maximize -(-8.21 + .71 / (1.09 - X(1) ^ 2))

minimize -.96 + .96 / (1.09 - X(2) ^ 2)

subject to

.3<= X(i) <= 1, for i=1, 2.

The problem above is based on the example in Miettinen, Eskelinen, Ruiz, and Luque [44, p. 430] and on the example in Narula and Weistroffer [51, p. 885].

0 DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ

86 M = -3E+50

89 EPSI1 = -5 + RND * 10

90 EPSI2 = -5 + RND * 10

91 EPSI3 = -5 + RND * 10

93 A(1) = .3 + RND * .7

94 A(2) = .3 + RND * .7

128 FOR I = 1 TO 3200

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

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

137 J = 1 + FIX(RND * 2)

138 GOTO 157

157 R = (1 - RND * 2) * A(J)

158 X(J) = A(J) + (RND ^ (RND * 15)) * R

161 GOTO 169

162 IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1

169 NEXT IPP

171 REM X(1) = INT(X(1))

178 IF X(1) > 1# THEN 1670

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

188 IF X(1) < .3# THEN 1670

189 IF X(2) < .3## THEN 1670

416 IF -(-2.6 - .03 * X(1) - .02 * X(2) - (.01 / (1.39 - X(1) ^ 2)) - (.3 / (1.39 - X(2) ^ 2))) < EPSI1 THEN 1670

418 IF -(-8.21 + .71 / (1.09 - X(1) ^ 2)) < EPSI2 THEN 1670

419 IF -(-4.07 - 2.27 * X(1)) < EPSI3 THEN 1670

447 PDU = .96 - .96 / (1.09 - X(2) ^ 2)

466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1471 gg01star = -(-2.6 - .03 * X(1) - .02 * X(2) - (.01 / (1.39 - X(1) ^ 2)) - (.3 / (1.39 - X(2) ^ 2)))

1476 gg02star = -(-8.21 + .71 / (1.09 - X(1) ^ 2))

1478 gg03star = -(-4.07 - 2.27 * X(1))

1557 GOTO 128

1670 NEXT I

1889 IF M < -999999999999999999999 THEN 1999

1912 REM IF M < 6.34 THEN 1999

1914 PRINT gg01star, gg02star, gg03star, JJJJ

1924 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [68]. The complete output of a single run through JJJJ= -31999 is shown below:

2.857959664670596 7.425714604430933 5.045617639378642

32000

.4297875063342034 .3 8.31583066296382D-17

-32000

2.861594885027973 7.335169314143761 5.26776520472646

31999

.5276498699235501 .3 8.31583066296382D-17

-31999

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 [68], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999 was 2 seconds, not including the time for “Creating .EXE file" (7 seconds, total, including the time for “Creating .EXE file").

**Acknowledgment**

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

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