Jsun Yui
Wong
Modeled on
the general-purpose nonlinear programming solver used many times for over a
decade in this blog, the computer program listed below attempts to solve the
following nonlinear program from Ecker and Kupferschmid [29, p. 366]:
Maximize
-2 * X(1) + X(1) ^ 2 + 3 * X(2) - 2 *
X(2) ^ 2 + 5 * X(3) - X(3) ^ 2
subject to
X(1) +
X(2) +X(3) <= 2
X(j) >= 0, j=1, 2, 3.
Because
there is only one long constraint, there is a reasonable chance that this long
constraint is binding. That is the
reason for using line 243, which is 243 X(1) = 2 - X(2)
- X(3).
This paper
uses the general-purpose nonlinear programming solver used in this blog many
times for over a decade to solve the problem above because “Continuous-state
dynamic programming problems can be quite complicated even when there is only
one state variable,” Ecker and Kupferschmid [29, p. 366] uses a dynamic
programming approach.
0
REM 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
118 FOR J44 = 1 TO 3
119 A(J44) = 0 + (RND * 7)
120 NEXT J44
128 FOR I = 1 TO 30000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 J = 1 + FIX(RND * 3)
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND *
15)) * r
161 GOTO 169
163 IF RND < .5 THEN X(J) = A(J)
- INT(RND * 15) ELSE X(J) = A(J) + INT(RND * 15)
169 NEXT IPP
172 REM X(1) = INT(X(1))
174 REM X(2) = INT(X(2))
176 REM X(3) = INT(X(3))
243 X(1) = 2 - X(2) - X(3)
326 FOR J44 = 1 TO 3
328 IF X(J44) < 0 THEN 1670
330 NEXT J44
477 P = -2 * X(1) + X(1) ^ 2 + 3 * X(2)
- 2 * X(2) ^ 2 + 5 * X(3) - X(3) ^ 2
478 REM
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 < 0 THEN 1999
1933 PRINT A(1), A(2), A(3), M, JJJJ
1999
NEXT JJJJ
This
computer program was run with qb64v1000-win [102]. Its complete output of one run through
JJJJ= -31994 is shown below:
0 .3815381 1.618462 6.326362 -32000
0 .3374368 1.662563 6.333283 -31998
0 .3537104 1.64629 6.32088 -31996
0 .340913 1.659087 6.333161 -31995
0 .3331461 1.666854 6.333333 -31994
Above there
is no rounding by hand; it is just straight copying by hand from the monitor
screen. By using the following computer
system and QB64v1000-win [102], the wall-clock time (not CPU time) for
obtaining the output through JJJJ = -31994 was one second, counting from
"Starting program...". One
can compare the computational procedure above with that in Ecker and
Kupferschmid [29, pp. 366-368].
The
computational results presented above were obtained from the following computer
system:
Processor:
Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz
2.26 GHz
Installed
memory (RAM): 4.00GB (3.87 GB usable)
System
type: 64-bit Operating System.
Acknowledgement
I would
like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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