Jsun Yui Wong
The reference K. E. Parsopoulos, M. N. Vrahatis, Particle swarm optimization for integer programming, IEEE 2002 should read E. C. Larskari, K. E. Parsopoulos, M. N. Vrahatis, Particle swarm optimization for integer programming, IEEE 2002.
Jsun Yui Wong
The computer program listed below seeks to solve the following test problem in Parsopoulos and Vrahatis [85, Test Problem 4]:
minimize f = (9 * X(1) ^ 2 + 2 * X(2) ^ 2 – 11) ^ 2 + (3 * X(1) + 4 * X(2) ^ 2 – 7) ^ 2
with solution at x*=(1, 1) and f(x*)=0, Parsopoulos and Vrahatis [85, Test Problem 4].
.
0 REM DEFDBL A-Z
1 REM 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)
12 FOR JJJJ = -32000 TO 32000 STEP .01
13 RANDOMIZE JJJJ
16 M = -1D+37
19 FOR J44 = 1 TO 2
21 A(J44) = -10 + FIX(RND * 21)
22 NEXT J44
128 FOR I = 1 TO 100
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO FIX(1 + RND ^ 5 * 5)
143 J = 1 + FIX(RND * 4)
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 R = (1 – RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 30)) * R
161 GOTO 165
162 REM IF RND < .16666 THEN X(J) = A(J) – FIX(1 + RND * 2.3) ELSE IF RND < .33333 THEN X(J) = A(J) + FIX(1 + RND * 2.3) ELSE IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * 20.3) ELSE IF RND < .66666 THEN X(J) = A(J) + FIX(1 + RND * 20.3) ELSE IF RND < .83333 THEN X(J) = A(J) – FIX(1 + RND * 200.3) ELSE X(J) = A(J) + FIX(1 + RND * 200.3)
164 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
165 NEXT IPP
166 FOR J44 = 1 TO 2
167 X(J44) = INT(X(J44))
168 NEXT J44
182 FOR J44 = 1 TO 2
183 IF X(J44) < -100 THEN 1670
184 IF X(J44) > 100 THEN 1670
185 NEXT J44
1020 P = -(9 * X(1) ^ 2 + 2 * X(2) ^ 2 – 11) ^ 2 – (3 * X(1) + 4 * X(2) ^ 2 – 7) ^ 2
1111 IF P <= M THEN 1670
1420 M = P
1444 FOR KLX = 1 TO 2
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1890 REM IF M < 30665 THEN 1999
1926 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [120]. The complete output of one run through JJJJ = -31999.84 is shown below:
1 -1 0 -32000
-1 1 -36 -31999.99
-1 1 -36 -31999.98
-1 1 -36 -31999.97
1 -1 0 -31999.96
-1 1 -36 -31999.95
1 -1 0 -31999.94
-1 -1 -36 -31999.93
-1 1 -36 -31999.92
-1 -1 -36 -31999.91
-1 -1 -36 -31999.9
1 -1 0 -31999.89
-1 1 -36 -31999.88
-1 1 -36 -31999.87
-1 -1 -36 -31999.86
-1 1 -36 -31999.85
1 1 0 -31999.84
Two distinct solutions are shown above at x*=(1, -1) and x*=(1, 1).
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz 3.00GHz, 4.00GB of RAM, and qb64v1000-win [120]. The wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.84 was 2 seconds, counting from “Starting program…”. One can compare the computational results above with those in Parsopoulos and Vrahatis [85, Test Problem 4].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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Jsun Yui Wong
The computer program listed below seeks to solve the following test problem in Parsopoulos and Vrahatis [85, Test Problem 6]:
minimize f = 2 * X(1) ^ 2 + 3 * X(2) ^ 2 + 4 * X(1) * X(2) – 6 * X(1) – 3 * X(2) ,
with solution at x*=(2, -1) and f*= -6, Parsopoulos and Vrahatis [85, Test Problem 6].
0 REM DEFDBL A-Z
1 REM 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)
12 FOR JJJJ = -32000 TO 32000 STEP .01
13 RANDOMIZE JJJJ
16 M = -1D+37
19 FOR J44 = 1 TO 2
21 A(J44) = -10 + FIX(RND * 21)
22 NEXT J44
128 FOR I = 1 TO 100
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO FIX(1 + RND ^ 5 * 5)
143 J = 1 + FIX(RND * 2)
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 R = (1 – RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 30)) * R
161 GOTO 165
162 REM IF RND < .16666 THEN X(J) = A(J) – FIX(1 + RND * 2.3) ELSE IF RND < .33333 THEN X(J) = A(J) + FIX(1 + RND * 2.3) ELSE IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * 20.3) ELSE IF RND < .66666 THEN X(J) = A(J) + FIX(1 + RND * 20.3) ELSE IF RND < .83333 THEN X(J) = A(J) – FIX(1 + RND * 200.3) ELSE X(J) = A(J) + FIX(1 + RND * 200.3)
164 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
165 NEXT IPP
166 FOR J44 = 1 TO 2
167 X(J44) = INT(X(J44))
168 NEXT J44
182 FOR J44 = 1 TO 2
183 IF X(J44) < -100 THEN 1670
184 IF X(J44) > 100 THEN 1670
185 NEXT J44
1016 P = -2 * X(1) ^ 2 – 3 * X(2) ^ 2 – 4 * X(1) * X(2) + 6 * X(1) + 3 * X(2)
1111 IF P <= M THEN 1670
1420 M = P
1444 FOR KLX = 1 TO 2
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1890 REM IF M < 30665 THEN 1999
1926 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [120]. The complete output of one run through JJJJ = -31999.9 is shown below:
2 -1 6 -32000
2 -1 6 -31999.99
2 -1 6 -31999.98
2 -1 6 -31999.97
3 -1 6 -31999.96
0 1 0 -31999.95
0 1 0 -31999.94
2 -1 6 -31999.93
4 -2 6 -31999.92
6 -4 0 -31999.91
3 -1 6 -31999.9
Three distinct solutions are shown above at x*=(2, -1), x*=(3, -1), and x*=(4, -2).
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz 3.00GHz, 4.00GB of RAM, and qb64v1000-win [120]. The wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.9 was 2 seconds, counting from “Starting program…”. One can compare the computational results above with those in Parsopoulos and Vrahatis [85, Test Problem 6].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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Jsun Yui Wong
The computer program listed below seeks to solve the following test problem in Parsopoulos and Vrahatis [84, Test Problem 1]:
minimize f = (X(1) -2)^2+(X(2)-1)^2
subject to
X(1)=2 * X(2) – 1,
X(1)^2/4+ X(2)^2 – 1 <= 0.
The best known solution is f*= 1.3934651, [84].
0 REM DEFDBL A-Z
1 REM 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)
12 FOR JJJJ = -32000 TO 32000 STEP .01
13 RANDOMIZE JJJJ
16 M = -1D+37
19 FOR J44 = 1 TO 2
21 A(J44) = -5 + RND * 10
22 NEXT J44
128 FOR I = 0 TO 10000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO FIX(1 + RND ^ 5 * 5)
143 j = 1 + FIX(RND * 5)
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 R = (1 – RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 30)) * R
161 GOTO 165
162 REM IF RND<.16666 THEN X(j)= …….
164 IF RND < .5 THEN X(j) = A(j) – FIX(1 + RND * .3) ELSE X(j) = A(j) + FIX(1 + RND * .3)
165 NEXT IPP
166 X(1) = 2 * X(2) – 1
167 IF X(1) ^ 2 / 4 + X(2) ^ 2 – 1 > 0 THEN 1670
1014 P = -(X(1) – 2) ^ 2 – (X(2) – 1) ^ 2
1111 IF P <= M THEN 1670
1420 M = P
1444 FOR KLX = 1 TO 2
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1890 REM IF M < 30665 THEN 1999
1926 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [120]. The complete output of one run through JJJJ = -31999.96 is shown below:
.8228756 .9114378 -1.393465 -32000
.8228756 .9114378 -1.393465 -31999.99
.8228756 .9114378 -1.393465 -31999.98
.8228756 .9114378 -1.393465 -31999.97
.8228756 .9114378 -1.393465 -31999.96
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz 3.00GHz, 4.00GB of RAM, and qb64v1000-win [120]. The wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.96 was 2 seconds, counting from “Starting program…”. One can compare the computational results above with those in Parsopoulos and Vrahatis [84, Test Problem 1, Table 1].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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Jsun Yui Wong
The computer program listed below seeks to solve the following test problem in Parsopoulos and Vrahatis [84, Test Problem 4]:
minimize f = 5.3578547 * X(3) ^ 2 + .8356891 * X(1) * X(5) + 37.293239 * X(1) – 40792.141
subject to
85.334407 + .0056858 * TONE + TTWO * X(1) * X(4) - .0022053 * X(3) * X(5) >=0,
85.334407 + .0056858 * TONE + TTWO * X(1) * X(4) - .0022053 * X(3) * X(5) <= 92,
80.51249 + .0071317 * X(2) * X(5) + .0029955 * X(1) * X(2) + .0021813 * X(3) ^ 2 >= 90,
80.51249 + .0071317 * X(2) * X(5) + .0029955 * X(1) * X(2) + .0021813 * X(3) ^ 2 <= 110,
9.300961 + .0047026 * X(3) * X(5) + .0012547 * X(1) * X(3) + .0019085 * X(3) * X(4) >= 20,
9.300961 + .0047026 * X(3) * X(5) + .0012547 * X(1) * X(3) + .0019085 * X(3) * X(4) <= 25,
78 <= X(1)<= 102, 33 <= X(2)<= 45, 27 <= X(i)<= 45, i=3, 4, 5,
where TONE = X(2) * X(5) and TTWO = .0006262,
-10 <= X(i) <= 10 where i=1 through 5.
The best known solution is f*= -30665.538.
One notes that the following computer program takes advantage of the “what if” question that (at least) one of the six long constraints is active. See line 170, which is 170 X(3) = (85.334407 + .0056858 * TONE + TTWO * X(1) * X(4) – 92) / (.0022053 * X(5)).
0 REM DEFDBL A-Z
1 REM 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)
12 FOR JJJJ = -32000 TO 32000 STEP .01
13 RANDOMIZE JJJJ
16 M = -1D+37
17 A(1) = 78 + RND * 24
18 A(2) = 33 + RND * 12
19 FOR J44 = 3 TO 5
21 A(J44) = 27 + RND * 18
22 NEXT J44
128 FOR I = 0 TO 3000000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO FIX(1 + RND ^ 5 * 5)
143 j = 1 + FIX(RND * 5)
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 R = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 30)) * R
161 GOTO 165
162 REM IF RND<.16666 THEN X(j)= .......
164 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * .3) ELSE X(j) = A(j) + FIX(1 + RND * .3)
165 NEXT IPP
169 TONE = X(2) * X(5): TTWO = .0006262
170 X(3) = (85.334407 + .0056858 * TONE + TTWO * X(1) * X(4) - 92) / (.0022053 * X(5))
171 IF 85.334407 + .0056858 * TONE + TTWO * X(1) * X(4) - .0022053 * X(3) * X(5) < 0## THEN 1670
173 IF 80.51249 + .0071317 * X(2) * X(5) + .0029955 * X(1) * X(2) + .0021813 * X(3) ^ 2 < 90## THEN 1670
174 IF 80.51249 + .0071317 * X(2) * X(5) + .0029955 * X(1) * X(2) + .0021813 * X(3) ^ 2 > 110## THEN 1670
175 IF 9.300961 + .0047026 * X(3) * X(5) + .0012547 * X(1) * X(3) + .0019085 * X(3) * X(4) < 20## THEN 1670
176 IF 9.300961 + .0047026 * X(3) * X(5) + .0012547 * X(1) * X(3) + .0019085 * X(3) * X(4) > 25## THEN 1670
177 IF X(1) < 78 THEN 1670
178 IF X(1) > 102 THEN 1670
179 IF X(2) < 33 THEN 1670
180 IF X(2) > 45 THEN 1670
182 FOR J44 = 3 TO 5
183 IF X(J44) < 27 THEN 1670
184 IF X(J44) > 45 THEN 1670
185 NEXT J44
1012 P = -5.3578547 * X(3) ^ 2 - .8356891 * X(1) * X(5) - 37.293239 * X(1) + 40792.141
1111 IF P <= M THEN 1670
1420 M = P
1444 FOR KLX = 1 TO 5
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1890 IF M < 30665 THEN 1999
1926 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [120]. The complete output of one run through JJJJ = -31999.96 is shown below:
78 33 29.99526 44.99902 36.77621
30665.51 -31000
78 33.00006 29.99529 44.99998 36.77573
30665.53 -31999.99
78 33.00011 29.99532 44.99996 36.77568
30665.53 -31999.98
78 33 29.99526 44.99972 36.77593
30665.53 -31999.97
78 33.00003 29.99527 44.99997 36.77579
30665.54 -31999.96
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz 3.00GHz, 4.00GB of RAM, and qb64v1000-win [120]. The wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.96 was 160 seconds, counting from “Starting program…”. One can compare the computational results above with those in Parsopoulos and Vrahatis [84, Test Problem 4, Table 1].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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Jsun Yui Wong
The computer program listed below seeks to solve the following test problem in Parsopoulos and Vrahatis [84, Test Problem 2]:
minimize f= (X(1) – 10) ^ 3 + (X(2) – 20) ^ 3
subject to
100 – (X(1) – 5) ^ 2 – (X(2) – 5) ^ 2 <= 0,
(X(1) – 6) ^ 2 + (X(2) – 5) ^ 2 – 82.81 <= 0
13 <= X(1) <=100,
0 <= X(2) <=100,
The best known solution is f*= -6961.81381.
0 REM DEFDBL A-Z
2 REM DEFINT K
3 DIM B(99), N(99), A(100255), H(99), L(99), U(99), X(100250), D(111), P(511), PS(33), J(30003), J44(30003), KKQQ(30003), KLX(30003), W(10111)
12 FOR JJJJ = -32000 TO 32000 STEP .01
13 RANDOMIZE JJJJ
16 M = -1D+37
21 A(1) = RND * 50
22 A(2) = RND * 50
128 FOR I = 1 TO 300000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO (1 + FIX(RND ^ 5 * 5))
148 J = 1 + FIX(RND * 2)
153 REM GOTO 162
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 REM
157 R = (1 – RND * 2) * (A(J))
160 X(J) = A(J) + (RND ^ (RND * 30)) * R
161 GOTO 165
162 REM IF RND < .16666 THEN X(J) = A(J) – FIX(1 + RND * 2.3) ELSE IF RND < .33333 THEN X(J) = A(J) + FIX(1 + RND * 2.3) ELSE IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * 20.3) ELSE IF RND < .66666 THEN X(J) = A(J) + FIX(1 + RND * 20.3) ELSE IF RND < .83333 THEN X(J) = A(J) – FIX(1 + RND * 200.3) ELSE X(J) = A(J) + FIX(1 + RND * 200.3)
164 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
165 NEXT IPP
175 IF 100 – (X(1) – 5) ^ 2 – (X(2) – 5) ^ 2 > 0## THEN 1670
177 IF (X(1) – 6) ^ 2 + (X(2) – 5) ^ 2 – 82.81 > 0## THEN 1670
197 IF X(1) < 13 THEN 1670
198 IF X(1) > 100 THEN 1670
199 IF X(2) < 0 THEN 1670
200 IF X(2) > 100 THEN 1670
959 P = -(X(1) – 10) ^ 3 – (X(2) – 20) ^ 3
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 2
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128
1670 NEXT I
1677 IF M < -9999 THEN 1999
1908 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [120]. The complete output of one run through JJJJ = -31999.66 is shown below:
14.09501 .8429765 6961.796 -31999.99
14.09501 .8429726 6961.801 -31999.83
14.09501 .8429765 6961.796 -31999.79
14.09501 .8429765 6961.796 -31999.77
14.09501 .8429706 6961.803 -31999.7
14.09501 .8429706 6961.803 -31999.67
14.095 .842969 6961.805 -31999.66
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz 3.00GHz, 4.00GB of RAM, and qb64v1000-win [120]. The wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.66 was 16 seconds, counting from “Starting program…”. One can compare the computational results above with those in Parsopoulos and Vrahatis [84, Test Problem 2, Table 1].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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A Test Problem for Constrained Optimization
Jsun Yui Wong
The computer program listed below seeks to solve the following test problem in Parsopoulos and Vrahatis [84, Test Problem 2]:
minimize f= (X(1) – 10) ^ 3 + (X(2) – 20) ^ 3
subject to
100 – (X(1) – 5) ^ 2 – (X(2) – 5) ^ 2 <= 0,
(X(1) – 6) ^ 2 + (X(2) – 5) ^ 2 – 82.81 <= 0
13 <= X(1) <=100,
0 <= X(2) <=100,
The best known solution is f*= -6961.81381.
0 REM DEFDBL A-Z
2 REM DEFINT K
3 DIM B(99), N(99), A(100255), H(99), L(99), U(99), X(100250), D(111), P(511), PS(33), J(30003), J44(30003), KKQQ(30003), KLX(30003), W(10111)
12 FOR JJJJ = -32000 TO 32000 STEP .01
13 RANDOMIZE JJJJ
16 M = -1D+37
21 A(1) = RND * 50
22 A(2) = RND * 50
128 FOR I = 1 TO 300000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO (1 + FIX(RND ^ 5 * 5))
148 J = 1 + FIX(RND * 2)
153 REM GOTO 162
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 REM
157 R = (1 – RND * 2) * (A(J))
160 X(J) = A(J) + (RND ^ (RND * 30)) * R
161 GOTO 165
162 REM IF RND < .16666 THEN X(J) = A(J) – FIX(1 + RND * 2.3) ELSE IF RND < .33333 THEN X(J) = A(J) + FIX(1 + RND * 2.3) ELSE IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * 20.3) ELSE IF RND < .66666 THEN X(J) = A(J) + FIX(1 + RND * 20.3) ELSE IF RND < .83333 THEN X(J) = A(J) – FIX(1 + RND * 200.3) ELSE X(J) = A(J) + FIX(1 + RND * 200.3)
164 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
165 NEXT IPP
175 IF 100 – (X(1) – 5) ^ 2 – (X(2) – 5) ^ 2 > 0## THEN 1670
177 IF (X(1) – 6) ^ 2 + (X(2) – 5) ^ 2 – 82.81 > 0## THEN 1670
197 IF X(1) < 13 THEN 1670
198 IF X(1) > 100 THEN 1670
199 IF X(2) < 0 THEN 1670
200 IF X(2) > 100 THEN 1670
959 P = -(X(1) – 10) ^ 3 – (X(2) – 20) ^ 3
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 2
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128
1670 NEXT I
1677 IF M < -9999 THEN 1999
1908 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [120]. The complete output of one run through JJJJ = -31999.66 is shown below:
14.09501 .8429765 6961.796 -31999.99
14.09501 .8429726 6961.801 -31999.83
14.09501 .8429765 6961.796 -31999.79
14.09501 .8429765 6961.796 -31999.77
14.09501 .8429706 6961.803 -31999.7
14.09501 .8429706 6961.803 -31999.67
14.095 .842969 6961.805 -31999.66
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz 3.00GHz, 4.00GB of RAM, and qb64v1000-win [120]. The wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.66 was 16 seconds, counting from “Starting program…”. One can compare the computational results above with those in Parsopoulos and Vrahatis [84, Test Problem 2, Table 1].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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Jsun Yui Wong
The computer program listed below seeks to solve the following test problem in the Wikipedia [121]:
minimize – (COS((X(1) – .1) * X(2))) ^ 2 – X(1) * SIN(3 * X(1) + X(2))
subject to
X(1) ^ 2 + X(2) ^ 2 < (2 * COS(t) – .5 * COS(2 * t) – .25 * COS(3 * t) – .125 * COS(4 * t)) ^ 2 + (2 * SIN(t)) ^ 2
where t = ATAN2(X(1), X(2)),
-2.25<= X(1) <=2.5,
-2.5<= X(2) <=1.75.
One line 154, whicn is 154 IF RND < 1.5 THEN GOTO 156 ELSE GOTO 162.
0 REM DEFDBL A-Z
2 REM DEFINT K
3 DIM B(99), N(99), A(100255), H(99), L(99), U(99), X(100250), D(111), P(511), PS(33), J(30003), J44(30003), KKQQ(30003), KLX(30003), W(10111)
12 FOR JJJJ = -32000 TO 32000 STEP .01
13 RANDOMIZE JJJJ
16 M = -1D+37
21 A(1) = RND * 2.5
22 A(2) = RND * 1.75
128 FOR I = 1 TO 300000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO (1 + FIX(RND ^ 5 * 5))
148 J = 1 + FIX(RND * 2)
153 REM GOTO 162
154 IF RND < 1.5 THEN GOTO 156 ELSE GOTO 162
156 REM
157 R = (1 – RND * 2) * (A(J))
160 X(J) = A(J) + (RND ^ (RND * 30)) * R
161 GOTO 165
162 REM IF RND < .16666 THEN X(J) = A(J) – FIX(1 + RND * 2.3) ELSE IF RND < .33333 THEN X(J) = A(J) + FIX(1 + RND * 2.3) ELSE IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * 20.3) ELSE IF RND < .66666 THEN X(J) = A(J) + FIX(1 + RND * 20.3) ELSE IF RND < .83333 THEN X(J) = A(J) – FIX(1 + RND * 200.3) ELSE X(J) = A(J) + FIX(1 + RND * 200.3)
164 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
165 NEXT IPP
167 t = ATAN2(X(1), X(2))
173 IF X(1) ^ 2 + X(2) ^ 2 – (2 * SIN(t)) ^ 2 – (2 * COS(t) – .5 * COS(2 * t) – .25 * COS(3 * t) – .125 * COS(4 * t)) ^ 2 >= 0## THEN 1670
197 IF X(1) < -2.25 THEN 1670
198 IF X(1) > 2.5 THEN 1670
199 IF X(2) < -2.5 THEN 1670
200 IF X(2) > 1.75 THEN 1670
956 P = (COS((X(1) – .1) * X(2))) ^ 2 + X(1) * SIN(3 * X(1) + X(2))
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 2
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128
1670 NEXT I
1677 IF M < -9999 THEN 1999
1908 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [120]. The complete output of one run through JJJJ = -31999.96 is shown below:
.6762341 2.412484E-07 1.606569 -32000
.6763346 3.818213E-08 1.606569 -31999.99
.6762227 2.046025E-07 1.606569 -31999.98
.6763284 1.086368E-08 1.606569 -31999.97
.6762912 2.447428E-07 1.606569 -31999.96
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz 3.00GHz, 4.00GB of RAM, and qb64v1000-win [120]. The wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.96 was 5 seconds, counting from “Starting program…”. One can compare the computational results above with those in the Wikipedia [121].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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Jsun Yui Wong
The computer program listed below seeks to solve the following test problem in the Wikipedia [121]:
minimize .1 * X(1) * X(2)
subject to
X(1) ^ 2 + X(2) ^ 2 <= (1 + .2 * COS(8 * arctan X(1) / X(2))) ^ 2
where -1.25<= X(1), X(2) <=1.25.
0 REM DEFDBL A-Z
2 REM DEFINT K
3 DIM B(99), N(99), A(100255), H(99), L(99), U(99), X(100250), D(111), P(511), PS(33), J(30003), J44(30003), KKQQ(30003), KLX(30003), W(10111)
12 FOR JJJJ = -32000 TO 32000 STEP .01
13 RANDOMIZE JJJJ
16 M = -1D+37
18 FOR J44 = 1 TO 2
21 A(J44) = -1 + RND * 2
25 NEXT J44
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO (1 + FIX(RND ^ 5 * 5))
148 J = 1 + FIX(RND * 2)
153 REM GOTO 162
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 REM
157 R = (1 – RND * 2) * (A(J))
160 X(J) = A(J) + (RND ^ (RND * 30)) * R
161 GOTO 165
162 REM IF RND < .16666 THEN X(J) = A(J) – FIX(1 + RND * 2.3) ELSE IF RND < .33333 THEN X(J) = A(J) + FIX(1 + RND * 2.3) ELSE IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * 20.3) ELSE IF RND < .66666 THEN X(J) = A(J) + FIX(1 + RND * 20.3) ELSE IF RND < .83333 THEN X(J) = A(J) – FIX(1 + RND * 200.3) ELSE X(J) = A(J) + FIX(1 + RND * 200.3)
164 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
165 NEXT IPP
168 REM
195 FOR J44 = 1 TO 2
197 IF X(J44) < -1.25 THEN 1670
198 IF X(J44) > 1.25 THEN 1670
204 NEXT J44
222 IF X(1) ^ 2 + X(2) ^ 2 – (1 + .2 * COS(8 * arctanX(1) / X(2))) ^ 2 > 0 THEN 1670
956 P = -.1 * X(1) * X(2)
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 2
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128
1670 NEXT I
1677 IF M < -9999 THEN 1999
1908 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [120]. The complete output of one run through JJJJ = -31999.97 is shown below:
.8564162 -.840566 7.198744E-02 -32000
-.8392708 .8576855 7.198304E-02 -31999.99
.8179137 -.8780758 7.181902E-02 -31999.98
-.8720713 .8243128 7.188596E-02 -31999.97
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz 3.00GHz, 4.00GB of RAM, and qb64v1000-win [120]. The wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.97 was 4 seconds, counting from “Starting program…”. One can compare the computational results above with those in the
Wikipedia [121].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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Jsun Yui Wong
The first constraint should read X(2) – X(1) ^ 2=0 instead of X(2) = X(1) ^ 2=0.
Jsun Yui Wong
The computer program listed below seeks to solve the following test problem G11 in Hedar and Fukushima [45, Problem G11, p. 23].
minimize X(1) ^ 2 + (X(2) – 1) ^ 2
subject to X(2) = X(1) ^ 2=0
-1<= x(i) <=1, i=1, 2,
0 REM DEFDBL A-Z
2 REM DEFINT K
3 DIM B(99), N(99), A(100255), H(99), L(99), U(99), X(100250), D(111), P(511), PS(33), J(30003), J44(30003), KKQQ(30003), KLX(30003), W(10111)
12 FOR JJJJ = -32000 TO 32000 STEP .01
13 RANDOMIZE JJJJ
16 M = -1D+37
18 FOR J44 = 1 TO 2
21 A(J44) = -1 + RND * 2
25 NEXT J44
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO (1 + FIX(RND ^ 5 * 5))
148 J = 1 + FIX(RND * 2)
153 REM GOTO 162
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 REM
157 R = (1 – RND * 2) * (A(J))
160 X(J) = A(J) + (RND ^ (RND * 30)) * R
161 GOTO 165
162 REM IF RND < .16666 THEN X(J) = A(J) – FIX(1 + RND * 2.3) ELSE IF RND < .33333 THEN X(J) = A(J) + FIX(1 + RND * 2.3) ELSE IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * 20.3) ELSE IF RND < .66666 THEN X(J) = A(J) + FIX(1 + RND * 20.3) ELSE IF RND < .83333 THEN X(J) = A(J) – FIX(1 + RND * 200.3) ELSE X(J) = A(J) + FIX(1 + RND * 200.3)
164 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
165 NEXT IPP
168 X(2) = X(1) ^ 2
195 FOR J44 = 1 TO 2
197 IF X(J44) < -1 THEN 1670
198 IF X(J44) > 1 THEN 1670
204 NEXT J44
944 P = -X(1) ^ 2 – (X(2) – 1) ^ 2
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 2
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128
1670 NEXT I
1677 IF M < -.76 THEN 1999
1908 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [120]. The complete output of one run through JJJJ = -31999.96 is shown below:
-.7070217 .4998796 -.75 -32000
.7071044 .4999966 -.75 -31999.99
.7071526 .5000648 -.75 -31999.98
.7070019 .4998516 -.75 -31999.97
-.707117 .5000144 -.75 -31999.96
Two distinct solutions are shown above.
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz 3.00GHz, 4.00GB of RAM, and qb64v1000-win [120]. The wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.96 was 3 seconds, counting from “Starting program…”. One can compare the computational results above with those in Hedar and Fukushima [45, Table 2, p. 15; 45, p. 23].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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