Addendum to: Solving a Complete System of Karush-Kuhn-Tucker Nonlinear Inequalities and Nonlinear Equations
Jsun Yui Wong
Line 174, which is 174 X(2) = ((8 * 2.718281828 ^ ((X(1) - 12) / 9) - 0 * X(9) + 4)), arises when one uses what if analysis and looks at X(3) as positive in the given constraint (X(3) * (8 * 2.718281828 ^ ((X(1) - 12) / 9) - X(2) + 4)) = 0. Then (8 * 2.718281828 ^ ((X(1) - 12) / 9) - X(2) + 4) = 0, which means X(2)= ((8 * 2.718281828 ^ ((X(1) - 12) / 9) + 4)).
Solving a Complete System of Karush-Kuhn-Tucker Nonlinear Inequalities and Nonlinear Equations
Jsun Yui Wong
The computer program listed below seeks to solve the following complete system of Karush-Kuhn-Tucker conditions from p. 305 of Ecker and Kupferschmid [29]:
4 * (X(1) - 20) ^ 3 + .8888888888888888 * X(3) * 2.7182818281828 ^ ((X(1) - 12) / 9) + 12 * X(4) * (X(1) - 12) -X(5) = 0,
4 * (X(2) - 12) ^ 3 - X(3)+ 25 * X(4) = 0,
8 * 2.718281828 ^ ((X(1) - 12) / 9) - X(2) + 4 <= 0,
6 * (X(1) - 12) ^ 2 + 25 * X(2) - 600 <= 0,
-X(1) + 12 <= 0,
(X(3) * (8 * 2.718281828 ^ ((X(1) - 12) / 9) - X(2) + 4)) = 0,
(X(4) * (6 * (X(1) - 12) ^ 2 + 25 * X(2) - 600)) = 0,
(X(5) * (-X(1) + 12)) = 0,
X(3), X(4), X(5) >= 0.
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 5
112 A(J44) = RND * 30
113 NEXT J44
128 FOR I = 1 TO 30000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 5)
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 * 5) ELSE X(B) = A(B) + FIX(RND * 5)
168 NEXT IPP
174 X(2) = ((8 * 2.718281828 ^ ((X(1) - 12) / 9) - 0 * X(9) + 4))
176 REM
178 X(3) = 4 * (X(2) - 12) ^ 3 + 25 * X(4)
179 X(5) = 4 * (X(1) - 20) ^ 3 + .8888888888888888 * X(3) * 2.7182818281828 ^ ((X(1) - 12) / 9) + 12 * X(4) * (X(1) - 12)
189 IF (8 * 2.718281828 ^ ((X(1) - 12) / 9) - X(2) + 4) > 0## THEN 1670
191 IF (6 * (X(1) - 12) ^ 2 + 25 * X(2) - 600) > 0## THEN 1670
199 IF -X(1) + 12 > 0## THEN 1670
261 FOR J44 = 3 TO 5
266 IF X(J44) < 0## THEN 1670
269 NEXT J44
339 PD1 = - ABS(X(4) * (6 * (X(1) - 12) ^ 2 + 25 * X(2) - 600)) - ABS(X(5) * (-X(1) + 12))
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 5
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1779 IF M < -.1 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [102]. Its output of one run through -31998 is shown below:
15.62949091119462 15.97376862886194 250.9965326157827
1.944834924692203D-27 1.541284322392756D-12 -5.594077439691267D-10
-32000
15.62949091119438 15.97376862886162 250.9965326157221
1.169933026788912D-29 91626494046344192D-12 -3.493927264787304D-11
-31998
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.
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Solvng a Nonlinear Program from the Literature
Jsun Yui Wong
The computer program listed below seeks to solve the following nonlinear programming problem from Ecker and Kupferschmid [29, p. 312]:
minimize X(1) ^ 2 + X(2) + X(3) ^ 2 + X(4)
subject to X(1) ^ 2 + X(2) + 4 * X(3) + 4 * X(4) - 4 = 0
-X(1) + X(2) + 2 * X(3) - 2 * X(4) ^ 2 + 2 = 0.
One notes line 122, which is 122 A(J44) = -5 + (RND * 10).
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
121 FOR J44 = 1 TO 4
122 A(J44) = -5 + (RND * 10)
123 NEXT J44
128 FOR I = 1 TO 35000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 4)
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) - 1 ELSE X(B) = A(B) + 1
168 NEXT IPP
177 X(2) = -X(1) ^ 2 - 4 * X(3) - 4 * X(4) + 4
344 PD1 = -X(1) ^ 2 - X(2) - X(4) - X(3) ^ 2 - 5000000 * ABS(-X(1) + X(2) + 2 * X(3) - 2 * X(4) ^ 2 + 2)
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1777 IF M < 1.611 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [102]. Its output of one run through JJJJ= -14557 is shown below:
-.5033886741597768 -4.84106550718483 1.538729565780501
.6081867716976217 1.611789899832981 -16217
-.5243840646265698 -4.876321629657315 1.543824578427412
.606511167178346 1.611437485946586 -14557
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 = -14557 was 13 minutes, counting from "Starting program...". One can compare the computational results above with those in Ecker an Kupferschmid [29, Table on p. 315].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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