Template for Solving Nonlinear Programming Problems with Equality Constraints
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following nonlinear programming problem from Ecker and Kupferschmidt [29, p. 304]:
Minimize (X(1) – 13 / 3) ^ 2 + (X(2) – 1 / 2) ^ 2 – X(3)
subject to
X(1) + (5 / 3) * X(2) -10=0,
(X(2) – 2) ^ 2 + X(3) -4=0.
One notes line 176 through line 348.
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 3
112 A(J44) = -50 + FIX(RND * 101)
113 NEXT J44
128 FOR I = 1 TO 20000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 2)
140 B = 1 + FIX(RND * 3)
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
176 X(1) = 10 - (5 / 3) * X(2)
178 X(3) = 4 - (X(2) - 2) ^ 2
348 PD1 = -(X(1) - 13 / 3) ^ 2 - (X(2) - 1 / 2) ^ 2 + X(3)
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 3
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1779 IF M < -99999 THEN 1999
1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT
This computer program was run with qb64v1000-win [102]. Its complete output of one run through -31998 is shown below:
5.8333333229285 2.5000000062429 3.7499999937571
-2.5 -32000
5.833333315594357 2.500000010643385 3.749999989356615
-2.5 -32000
5.833333333450572 2.499999999929657 3.750000000070343
-2.5 -32000
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 = -31998 was 2 seconds, counting from “Starting program…”.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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