Jsun Yui Wong
I. (X(1) - 2) ^ 2 + (X(2) - 3) ^ 2<=1
The computer program listed below seeks to solve the following problem:
Minimize (5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4)
subject to
(X(1) - 2) ^ 2 + (X(2) - 3) ^ 2<=1,
0<=X(1) <= 3,
0<=X(2) <= 3,
X(1) and X(2) are integer variables.
The problem above is Example 5 in Shen, Zhang, and Wang [32, p. 12 of 16] plus the integer restriction.
X(3) below is an added slack variable.
One notes line 193 and line 194, which are 193 X(1)=INT(X(1)) and 194 X(2)=INT(X(2)), respectively.
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
72 A(1) = RND * 3
75 A(2) = RND * 3
128 FOR I = 1 TO 2000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))
181 J = 1 + FIX(RND * 2)
183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r
191 NEXT IPP
193 X(1)=INT(X(1))
194 X(2)=INT(X(2))
201 IF X(1) < 0 THEN 1670
203 IF X(1) > 3 THEN 1670
211 IF X(2) < 0 THEN 1670
213 IF X(2) > 3 THEN 1670
311 X(3) = 1 - (X(1) - 2) ^ 2 - (X(2) - 3) ^ 2
333 FOR J44 = 3 TO 3
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
339 NEXT J44
378 POBA = -(5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4) + 1000000 * (X(3))
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 3
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -111111 THEN 1999
1900 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [40]. The complete output through
JJJJ = -31999.4600000001 is shown below:
2 3 0 -78246 -31999.80000000003
2 3 0 -78246 -31999.73000000004
2 3 0 -78246 -31999.58000000007
2 3 0 -78246 -31999.53000000008
2 3 0 -78246 -31999.51000000008
2 3 0 -78246 -31999.4600000001
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 [40], the wall-clock time for obtaining the output through
JJJJ= -31999.4600000001 was about 2 seconds, not including the time for "Creating .EXE file."
II. (X(1) - 2) ^ 2 + (X(2) - 2) ^ 2<=1
The computer program listed below seeks to solve the following problem:
Minimize (5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4)
subject to
(X(1) - 2) ^ 2 + (X(2) - 2) ^ 2<=1,
0<=X(1) <= 3,
0<=X(2) <= 3,
X(1) and X(2) are integer variables.
The problem above is Example 5.2 in Shao and Ehrgott [29, p. 723] plus the integer restriction.
X(3) below is an added slack variable.
One notes line 193 and line 194, which are 193 X(1)=INT(X(1)) and 194 X(2)=INT(X(2)), respectively.
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
72 A(1) = RND * 3
75 A(2) = RND * 3
128 FOR I = 1 TO 2000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))
181 J = 1 + FIX(RND * 2)
183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r
191 NEXT IPP
193 X(1)=INT(X(1))
194 X(2)=INT(X(2))
201 IF X(1) < 0 THEN 1670
203 IF X(1) > 3 THEN 1670
211 IF X(2) < 0 THEN 1670
213 IF X(2) > 3 THEN 1670
311 X(3) = 1 - (X(1) - 2) ^ 2 - (X(2) - 2) ^ 2
333 FOR J44 = 3 TO 3
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
339 NEXT J44
378 POBA = -(5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4) + 1000000 * (X(3))
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 3
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -111111 THEN 1999
1900 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [40]. The complete output through JJJJ = -31999.33000000011 is shown below:
2 3 0 -78246 -31999.80000000003
2 3 0 -78246 -31999.33000000011
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 [40], the wall-clock time for obtaining the output through
JJJJ= -31999.33000000011 was about 2 seconds, not including the time for "Creating .EXE file."
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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