Jsun Yui Wong
The computer program listed below seeks to solve the following 0-1 fractional program from Kapoor and Arora [38, p. 201, Example 2]:
Minimize (2 - 2 * X(1) ^ 2 + X(2) ^ 2 - X(3) ^ 2 + 2 * X(1) * X(2) + 4 * X(2) * X(3)) / (3 - X(1) ^ 2 - X(2) ^ 2 + X(3) ^ 2 + 6 * X(1) * X(3))
subject to
X(1) + X(2) + X(3) >= 2
2 * X(2) + X(3) >= 2
.5 * X(1) + X(2) >= 1
X(1), X(2), X(3)=0 or 1.
One notes line 128, which is 128 FOR I = 1 TO 2.
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
92 A(1) = FIX(RND * 2)
93 A(2) = FIX(RND * 2)
96 A(3) = FIX(RND * 2)
128 FOR I = 1 TO 2
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 2)
153 J = 1 + FIX(RND * 3)
155 GOTO 164
156 r = (1 - RND * 2) * A(J)
159 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
162 IF RND < .5 THEN X(J) = A(J) - INT(RND * 2) ELSE X(J) = A(J) + INT(RND * 2)
163 GOTO 169
164 IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
222 IF X(1) + 2 * X(2) + X(3) < 2 THEN 1670
225 IF 2 * X(1) + 3 * X(2) - 4 * X(3) > 5 THEN 1670
324 IF .5 * X(1) + X(3) < 1 THEN 1670
511 PDU = -(2 - 2 * X(1) ^ 2 + X(2) ^ 2 - X(3) ^ 2 + 2 * X(1) * X(2) + 4 * X(2) * X(3)) / (3 - X(1) ^ 2 - X(2) ^ 2 + X(3) ^ 2 + 6 * X(1) * X(3))
666 P = PDU
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 3
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -900000 THEN 1999
1956 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [72]. The complete output of a single run through JJJJ= -31991 is shown below:
0 1 1 -2 -32000
1 0 1 .1111111111111111
-31998
1 0 1 .1111111111111111
-31996
0 1 1 -2 -31995
1 0 1 .1111111111111111
-31994
1 1 1 -.75 -31993
1 1 1 -.75 -31991
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 [72], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31991 was 1 or 2 seconds, not including the time for “Creating .EXE file" (7 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Kapoor and Arora [38, p. 207].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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