Solving Brown’s Almost Linear System in 13000 General Integer Variables and 13000 Equations, Edition with Less Cold Starts
Jsun Yui Wong
Using the integer programming algorithm here the computer program below aims to solve the 13000 integer variables and 13000 equations system based on the immediately following much smaller 3 variables and 3 equations system in Sotiropoulos, Nikas, and Grapsa [101]:
2 * X(1) + X(2) + X(3) – 4=0,
X(1) + 2 * X(2) + X(3) – 4=0,
X(1) * X(2) * X(3) – 1=0.
One notes line 257, which is 257 IF ABS(X(J44)) > 1 THEN 1670.
Whereas line 112 of the earlier edition is 112 A(J44) = -1 + FIX(RND * 2), here line 112 is 112 A(J44) = -1 + FIX(RND * 3). The latter generates less cold starts.
0 DEFDBL A-Z
1 REM DEFINT I, J, K, A, X
2 DIM B(99), N(99), A(13002), H(99), L(99), U(99), X(13002), D(111), P(111), J(99), LHS(33)
5 DIM AA(99), HR(32), HHR(32), PLHS(44), LB(22), UB(22), PX(44), J44(44), PS(13122), PS1(13111), SUMM(13111)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 13000
112 A(J44) = -1 + FIX(RND * 3)
117 NEXT J44
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 13000
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 5)
148 J = 1 + FIX(RND * 13000)
154 IF RND < .5 THEN GOTO 157 ELSE GOTO 164
157 r = (1 – RND * 2) * (A(J))
160 X(J) = A(J) + (RND ^ (RND * 30)) * r
161 GOTO 166
164 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
166 NEXT IPP
251 FOR J44 = 1 TO 13000
254 X(J44) = INT(X(J44))
257 IF ABS(X(J44)) > 1 THEN 1670
259 NEXT J44
311 CONS = 0
315 FOR J44 = 1 TO 13000
318 CONS = CONS + X(J44)
319 NEXT J44
341 PROD = 1
344 FOR J44 = 1 TO 13000
347 PROD = PROD * X(J44)
349 NEXT J44
361 FOR J44 = 1 TO 12999
364 SUMM(J44) = CONS + X(J44)
369 NEXT J44
381 PS = 0
383 FOR J44 = 1 TO 12999
386 PS = PS – ABS(SUMM(J44) – 13001)
389 NEXT J44
943 PD1 = PS – ABS(PROD – 1)
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 13000
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M < -9 THEN 1999
1906 PRINT A(1), A(2), A(3), A(4), A(5), A(12996), A(12997), A(12998), A(12999), A(13000), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [121]. The complete output of one run through JJJJ= -31998 is shown below:
1 1 1 1 1
1 1 1 1 1
0 -32000
1 1 1 1 1
1 1 1 1 1
-26001 -31999
1 1 1 1 1
1 1 1 1 1
0 -31998
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
One notes that only 10 values of the 13000 values of the 13000 variables are shown above in accordance with line 1906, which is 1906 PRINT A(1), A(2), A(3), A(4), A(5), A(12996), A(12997), A(12998), A(12999), A(13000), M, JJJJ.
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 [121]. The wall-clock time (not CPU time) for obtaining the output shown above was 1 hour and 45 minutes, counting from “Starting program…”.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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computerprogramsandresults 