Solving in General Integers a Nonlinear System of Five Simultaneous Nonlinear Equations

Jsun Yui Wong

The computer program listed below seeks one integer solution to the following given system of five nonlinear equations:

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.

Equations 2, 3, and 4 above are based on page 112 of Hock and Schittkowski [6]. The last is based on page 117 of Hock and Schittkowski [6].

One notes line 94, which is 94 A(KK) = 1 + FIX(RND * 10).  

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
5 FOR JJJJ = -32000 TO 32000
    14 RANDOMIZE JJJJ
    16 M = -1D+17
    91 FOR KK = 1 TO 13
        94 A(KK) = 1 + FIX(RND * 10)

    95 NEXT KK
    128 FOR I = 1 TO 10000000

        129 FOR K = 1 TO 13
            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 13)
            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
        265 FOR J45 = 1 TO 13
            266 IF X(J45) < 1 THEN X(J45) = A(J45)
        267 NEXT J45
        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


        277 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15)

        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 13
            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
    1670 NEXT I
    1888 IF M < -3 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), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [13]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31976 is shown below:

6      4      11      2      6
4      11      5       9      1
1      1      3      -2.742824      -32000    
...
6      10      9      9      7
6      3      6      6      1
1      1      1      -1.651043      -31998

7      7      10      7      9
7      6      7       1      1
1      1      1      -2.003829       -31997

12      5      6     7     6  
10      6      1     2     4
1      2       4     -2 406149     -31989

9      2      11      1      6
3      11     6      2      5
1      2      5      -1.766955    -31984

6     4      2      7     9
7     5      4      5     7
1     9      6      -.3914129      -31983

6      6      6      6      6
6      6      6      1      9
8      5      3      -9.020425E-04
-31976

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 [13], the wall-clock time through JJJJ=-31976 was 27 minutes.

Acknowledgment

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

References

[1] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.

[2] F. Glover, 1986. Future Paths for Integer Programming and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549

[3] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.

[4] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.

[5] 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.

[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.

[7] 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.

[8] 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.

[9] 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.

[10] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.

[11] 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.

[12] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.

[13] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[14] Wolfram Research, Inc., Diophantine Polynomial Systems.
https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.

[15] 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.

[16] 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/

ERRATUM: The General Mixed Integer Nonlinear Programming (MINLP) Computer Program/Solver Solving Simultaneously a Nonlinear System of Two Simultaneous Nonlinear Equations Involving 100,000 General Integer Variables

Jsun Yui Wong

The last paragraph should read as follows:

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [12], the wall-clock time for reaching M=0 was 6 hours.

The General Mixed Integer Nonlinear Programming (MINLP) Computer Program/Solver Solving Simultaneously a Nonlinear System of Two Simultaneous Nonlinear Equations Involving 100,000 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks one integer solution to the following problem:

X(1) * X(2) * X(3) * X(4) * ...* X(100000) - 1 = 0,

99999
sigma [ 100 * (X(k + 1) - X(k) ^ 2) ^ 2 + (1 - X(k)) ^ 2 ] = 0,
k=1

and each unknown = 0, 1, or 2.

The first equation above is a part of the Brown almost linear function [1, p. 660]. The second is based on the Rosenbrock function in Schitkowski [11, pp. 118-123].

One notes line 88, line 94, line 189, line 195, and line 196.


0 REM DEFDBL A-Z

3 DEFINT X

4 DIM X(100042), A(100042), K(100033)


5 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ
    16 M = -1D+57


    88 A(1) = 2

    91 FOR KK = 2 TO 100000


        94 A(KK) = 1



    95 NEXT KK

    128 FOR I = 1 TO 30000000


        129 FOR K = 1 TO 100000


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

            171 REM B = 1 + FIX(RND * 100003)


            175 REM  GOTO 190


            181 B = 1 + FIX(RND * 100003)

            182 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) - 1 ELSE X(B) = A(B) + 1


            190 REM IF A(B) = 0 THEN X(B) = 1 ELSE X(B) = 0


        191 NEXT IPP

        193 REM GOTO 201


        194 FOR J49 = 1 TO 100000
            195 IF X(J49) < 0 THEN GOTO 1670

            196 IF X(J49) > 2 THEN GOTO 1670


        197 NEXT J49


        201 PRODBROWN = 1
        203 FOR J33 = 1 TO 100000

            206 PRODBROWN = PRODBROWN * X(J33)


        209 NEXT J33


        222 N1 = PRODBROWN - 1
        225 GOTO 257


        231 SUMSCHI = 0
        236 FOR J22 = 1 TO 100000

            239 SUMSCHI = SUMSCHI + X(J22) ^ 2


        241 NEXT J22
        244 N9 = -SUMSCHI + 100000



        246 REM N5 = -(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(1) ^ 4 + X(2) ^ 4 + X(3) ^ 4 + X(4) ^ 4 + X(5) ^ 4 + X(6) ^ 4 + X(7) ^ 4 + X(8) ^ 4) + (X(1) ^ 3 + X(2) ^ 3 + X(3) ^ 3 + X(4) ^ 3 + X(5) ^ 3 + X(6) ^ 3 + X(7) ^ 3 + X(8) ^ 3) ^ 2



        257 sumrose = 0
        259 FOR j44 = 1 TO 99999

            261 sumrose = sumrose + 100 * (X(j44 + 1) - X(j44) ^ 2) ^ 2 + (1 - X(j44)) ^ 2
        263 NEXT j44
        265 N6 = sumrose - 0

        277 P = -ABS(N1) - ABS(N6)



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 100000

            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1666 REM PRINT A(1), A(49999), A(50000), M, JJJJ


        1667 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(99994), A(99995), A(99996), A(99997), A(99998), A(99999), A(100000), M, JJJJ


    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), A(29996), A(29997), A(49998), A(49999), A(50000)

    1912 PRINT A(1111), A(1112), A(1113), A(1114), A(1115), A(3336), A(3337), A(3338), A(3339), A(3340), A(5996), A(5997), A(25998), A(25999), A(50000), M, JJJJ

1999 NEXT JJJJ


This computer program was run with qb64v1000-win [12]. Copied by hand from the screen, the computer program’s output through M=0 is shown below:

2      1      1      1      1
1      1      1      1      1
1      1      1      1      1
-2910      -32000

1      1      1      1      1
1      1      1      1      1
1      1      1      1      1
-2005      -32000

1      1      1      1      1
1      1      1      1      1
1      1      1      1      1
-1203      -32000

1      1      1      1      1
1      1      1      1      1
1      1      1      1      1
-403      -32000

1      1      1      1      1
1      1      1      1      1
1      1      1      1      1
-202      -32000

1      1      1      1      1
1      1      1      1      1
1      1      1      1      1
0      -32000

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

Of the 100,000 A's, only the 15 A's of line 1667 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [12], the wall-clock time for reaching M=0 was 10 hours.

Acknowledgment

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

References

[1] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[2] F. Glover, 1986. Future Paths for Integer Programming and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[3] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[4] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[5] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. www.SciRP.org/journal/am.
[6] 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.
[7] 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.
[8] 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.
[9] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[10] 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.
[11] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[12] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[13] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[14] 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.
[15] 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/

Solving in General Integers a Nonlinear System of Two Simultaneous Nonlinear Equations with Cold Starts A(KK) = -300000 + FIX(RND * 600001)

Jsun Yui Wong

The computer program listed below seeks one or more integer solutions to the following given system of two nonlinear equations:

 (X(1) - 1) ^ 2 + (X(1) - X(2)) ^ 2 + (X(3) - 1) ^ 2 + (X(4) - 1) ^ 4 + (X(5) - 1) ^ 6  =   4

 (X(1) - 1) ^ 2 + (X(1) - X(2)) ^ 2 + (X(2) - X(3)) ^ 2 + (X(3) - X(4)) ^ 4 + (X(4) - X(5)) ^ 4  =   1.

The two equation are based on page 97 and page 99 of Hock and Schittkowski [6], respectively.
One notes the starting vectors of line 94, which is 94 A(KK) = -300000 + FIX(RND * 600001).

0 REM DEFDBL A-Z

3 DEFINT X

4 DIM X(100042), A(100042), K(100033)


5 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ
    16 M = -1D+57

    91 FOR KK = 1 TO 5


        94 A(KK) = -300000 + FIX(RND * 600001)


    95 NEXT KK

    128 FOR I = 1 TO 1000


        129 FOR K = 1 TO 5


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


            181 B = 1 + FIX(RND * 5)

            182 IF RND < .5 THEN 183 ELSE GOTO 189


            183 R = (1 - RND * 2) * A(B)



            186 X(B) = A(B) + (RND ^ 3) * R

            188 GOTO 191

            189 IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

            190 REM IF A(B) = 0 THEN X(B) = 1 ELSE X(B) = 0



        191 NEXT IPP


        224 N1 = (X(1) - 1) ^ 2 + (X(1) - X(2)) ^ 2 + (X(3) - 1) ^ 2 + (X(4) - 1) ^ 4 + (X(5) - 1) ^ 6 - 4

        276 N2 = (X(1) - 1) ^ 2 + (X(1) - X(2)) ^ 2 + (X(2) - X(3)) ^ 2 + (X(3) - X(4)) ^ 4 + (X(4) - X(5)) ^ 4 - 1



        277 P = -ABS(N1) - ABS(N2)



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 5

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

    1670 NEXT I
    1888 IF M < -10 THEN 1999


    1910 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [13]. Copied by hand from the screen, the computer program’s output through JJJJ=-27005 is summarized below:
.
.
.
0      0      0      0      1
-2      -27070
0      0      0      0      1
-2      -27058
0      0      0      0      0
0      -27046
0      0      0      0      0
0      -27037
2      2      2      2      2
0      -27036
1      0      0      0      0
0      -27033
0      0      0      0      0
0      -27019
0      -1      0      0      1
-3      -27016
1      2      2      2      2
0      -27013
0      0      0      0      0
0      -27006
1      0      0      0      0
0      -27005

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 [13], the wall-clock time through JJJJ=-27005 was 20 seconds.

Acknowledgment

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

References
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