Nonlinear Integer/Continuous/Discrete Programming Solver: Simultaneously Solving in General Integers a Nonlinear System of 18 Simultaneous Nonlinear Equations in 36 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following problem:

- X(35) * (X(35) + 1) + 2 * X(36) = 18,

(X(35) - 1) ^ 2 + (X(36) - 6) ^ 2 = 25

EXP(X(32) * X(33) * X(34)) = 1,

3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3 = 0,

3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3 = 6666.6666666,

(X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27) = 714,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2 = 3155,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 = 3119,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2
= 2902,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 = 1590,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 = 659,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 = 395,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 1 + X(11) ^ 1 + X(12) ^ 1 + X(13) ^ 1 = 348,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 = 468,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2205.868,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2* X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2206.889,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2208.886,

-.5 * (X(1) * X(4) – X(2) * X(3) + X(3) * X(9) – X(5) * X(9) + X(5) * X(8) – X(6) * X(7)) = 0,

and

-5<= Integers X(i)<= 15 for i=1 through 36 except 0<= Integer X(13)<= 10.

The first two equations come from Burden, Faires, and Burden [2, page 648]. Equations 3 through 6 are based on page 100, page 118, 116, and page 111 of Hock and Schittkowski [8], respectively. Equations 7 through 11 are based on page 147 of Asaadi [1]. Equation 12 is based on page 122 of Hock and Schittkowski [8], equations 15, and 16, and 17 are based on page 112 of the same book, and equation 18 is based on page 117 of the same book, as well.

One notes line 13, which is 13 A(KNEW) = -5 + FIX(RND * 20.98).

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
13 A(KNEW) = -5 + FIX(RND * 20.98)

14 RANDOMIZE JJJJ

16 M = -1D+317

22 REM

91 FOR KK = 1 TO 36

94 A(KK) = A(KNEW)

95 NEXT KK
128 FOR I = 1 TO 3000000

129 FOR K = 1 TO 36

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 36)

182 IF RND < -.1 THEN 183 ELSE GOTO 189

183 R = (1 - RND * 2) * A(B)
186 X(B) = A(B) + (RND ^ 3) * R
187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
188 GOTO 191
189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99)
191 NEXT IPP
255 FOR J45 = 1 TO 36

256 IF X(J45) < -5 THEN X(J45) = A(J45)

257 IF X(J45) > 15 THEN X(J45) = A(J45)

259 NEXT J45

264 X(13) = 348 - (X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 1 + X(11) ^ 1 + X(12) ^ 1)

266 IF X(13) < 0 THEN 1670

267 IF X(13) > 10 THEN 1670

268 N11 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2205.868
269 N12 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2206.889
270 N13 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2208.886
273 N14 = -.5 * (X(1) * X(4) - X(2) * X(3) + X(3) * X(9) - X(5) * X(9) + X(5) * X(8) - X(6) * X(7)) - 0

275 N15 = X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 - 468

276 N16 = -395 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2

278 N17 = -659 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2
279 N18 = -1590 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2

280 N19 = -2902 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2

281 N20 = -3119 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2

282 N21 = -3155 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2

283 N22 = -714 + (X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)

284 N23 = -6666.6666666 + 3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3

291 N24 = 0 + 3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3

295 IF (X(32) * X(33) * X(34)) > 80 THEN GOTO 1670

296 N25 = -1 + EXP(X(32) * X(33) * X(34))

298 N26 = -18 - X(35) * (X(35) + 1) + 2 * X(36)

299 N27 = -25 + (X(35) - 1) ^ 2 + (X(36) - 6) ^ 2

1287 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15) - ABS(N16) - ABS(N17) - ABS(N18) - ABS(N19) - ABS(N20) - ABS(N21) - ABS(N22) - ABS(N23) - ABS(N24) - ABS(N25) - ABS(N26) - ABS(N27)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 36

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P

1670 NEXT I
1888 IF M < -100 THEN 1999

1910 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10)
1912 PRINT A(11), A(12), A(13), A(14), A(15), A(16), A(17), A(18), A(19), A(20)

1914 PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), A(28), A(29), A(30)

1915 PRINT A(31), A(32), A(33), A(34), A(35), A(36), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [15]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31920 is shown below:

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 0 6
10 10 0 14 1
6 5 1 1 0
0 -4 0 1 -2
10 -9.020425E-04 -31995

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 7 0 15 7
12 9 0 10 0
11 3 1 1 0
0 0 7 -2 -2
10 -22.0009 -31993

6 6 6 6 6
6 6 6 6 7
5 6 6 3 5
6 7 6 0 6
12 4 4 6 1
4 1 1 1 0
0 0 -1 -5 -2
10 -18.0009 -31988

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
11 11 2 8 2
2 -1 1 1 0
0 8 2 0 -2
2 -16.0009 -31982

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 7 -5
11 15 5 11 -1
3 0 1 1 0
0 0 6 7 -2
10 -11.0009 -31937

6 6 6 6 6
6 6 6 6 6
5 6 7 1 4
6 7 6 0 6
6 2 -1 4 -1
4 2 1 1 0
0 -4 3 0 -2
10 -17.0009 -31925

6 6 6 6 6
6 6 6 6 6
6 6 6 3 5
6 7 5 13 6
4 6 -4 4 -1
5 2 1 1 0
0 6 -4 0 -2
10 -18.0009 -31920

Above there is no rounding by hand; it is just straight copying by hand from the screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [15], the wall-clock time through JJJJ=-31920 was 20 minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References
[1] J. Asaadi, A Computational Comparison of Some Nonlinear Programs, Mathematical Programming, Vol. 4 (1973), pp. 144-154.
[2] R. Burden, J. Faires, and A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[4] F. Glover, 1986. Future Paths for Integer ng and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[5] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[6] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[7] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. http://www.SciRP.org/journal/am.
[8] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[9] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.
[10] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[11] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[13] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[16] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[17] Jsun Yui Wong (2013, November 11). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html.
[18] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/

Nonlinear Integer/Continuous/Discrete Programming Solver

Saturday, December 3, 2016

Simultaneously Solving in General Integers a Nonlinear System of 18 Simultaneous Nonlinear Equations in 36 General Integer Variables

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