September 26, 2015 by 123writereadmath

Testing the Nonlinear Integer Programming Solver with Another Diophantine Equation Involving a Sum of Like Powers

Jsun Yui Wong

The computer program listed below seeks to solve the following Diophantine equation from Reference 14 of the case of k=4 and n=4:

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

The following computer program uses qb64v1000-win [12, 15]. One notes line 88 below, which is 88 FOR JJJJ = -32000 TO 32000 STEP .1.

0 DEFDBL A-Z
1 DEFINT I, K, A, X

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), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
88 FOR JJJJ = -32000 TO 32000 STEP .1

89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 6

112 A(J44) = 20 + (RND * 500)

113 NEXT J44
128 FOR I = 1 TO 5000

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)
150 R = (1 – RND * 2) * A(B)
155 IF RND < .5 THEN 160 ELSE GOTO 167
160 X(B) = (A(B) + RND ^ 3 * R)
164 REM IF RND<.5 THEN X(B)=(A(B)-RND*10) ELSE X(B)=(A(B) +RND*10)
165 GOTO 168
167 IF RND < .5 THEN X(B) = CINT(A(B) – 1) ELSE X(B) = CINT(A(B) + 1)
168 NEXT IPP
169 REM GOTO 185
171 IF X(1) = X(2) THEN 1670
172 IF X(1) = X(3) THEN 1670
173 IF X(2) = X(3) THEN 1670
185 FOR J44 = 1 TO 6

186 IF X(J44) > 1000 THEN X(J44) = 330
187 IF X(J44) < 1 THEN X(J44) = 20
188 NEXT J44
195 X(5) = ((X(1) ^ 4# + X(2) ^ 4# + X(3) ^ 4# + X(4) ^ 4#)) ^ .25##
215 N(7) = X(5) ^ 4# – X(1) ^ 4# – X(2) ^ 4# – X(3) ^ 4# – X(4) ^ 4#
322 PD1 = -ABS(N(7))
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 5
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 0 THEN 1999

1904 PRINT A(1), A(2), A(3), A(4)
1905 PRINT A(5), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [12, 15]. Copied by hand from the screen, the computer program’s complete output through
JJJJ=32000 is shown below:

272 30 315 120
353 0 22318.59999977054

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

The solution shown above was found in 1911 by R. Norrie–see Reference 14.

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, 15], the wall-clock time for obtaining the output through JJJJ=32000 was about four hours.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[4] Martin Gardner (1983), Diophantine Analysis and Fermat’s Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). http://www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[5] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler’s Conjecture on Sums of Like Powers. The Bulletin of American Mathematical Society, Vol. 72, 1966, page 1079.

[6] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler’s Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[7] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[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] 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.

[10] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[11] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[12] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview

[13] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[14] Wikipedia, Euler’s sum of powers conjecture. https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture

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

September 24, 2015 by 123writereadmath

Testing the Nonlinear Integer Programming Solver with Lander and Parkin’s Counterexample to Euler’s Conjecture on Sums of Like Powers, QB64 Edition

Jsun Yui Wong

Based on the computer program in [16], the computer program below seeks to solve the following Diophantine equation:
X(1)^5+X(2)^5+X(3)^5+X(4)^5 = X(5)^5, which is based on the following equation taken from Lander and Parkin [5, p. 1079] and Lander and Parkin [6, p. 102]: 27^5+84^5+110^5+133^5= 144^5.

The following computer program uses qb64v1000-win [12, 15]. One notes line 88 below, which is 88 FOR JJJJ = -32000 TO 32000 STEP .5.

0 DEFDBL A-Z
1 DEFINT I, K, A, X

89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 5
112 A(J44) = 5 + (RND * 200)
113 NEXT J44
128 FOR I = 1 TO 350
129 FOR KKQQ = 1 TO 8
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 8)
150 R = (1 – RND * 2) * A(B)
155 IF RND < .5 THEN 160 ELSE GOTO 167
160 X(B) = (A(B) + RND ^ 3 * R)
164 REM IF RND<.5 THEN X(B)=(A(B)-RND*10) ELSE X(B)=(A(B) +RND*10)
165 GOTO 168
167 IF RND < .5 THEN X(B) = CINT(A(B) – 1) ELSE X(B) = CINT(A(B) + 1)
168 NEXT IPP
169 REM GOTO 185
171 IF X(1) = X(2) THEN 1670
172 IF X(1) = X(3) THEN 1670
173 IF X(2) = X(3) THEN 1670
185 FOR J44 = 1 TO 5
186 IF X(J44) > 1000 THEN X(J44) = 50
187 IF X(J44) < 10 THEN X(J44) = 50
188 NEXT J44
195 X(5) = ((X(1) ^ 5# + X(2) ^ 5# + X(3) ^ 5# + X(4) ^ 5#)) ^ .2#
215 N(7) = X(5) ^ 5# – X(1) ^ 5# – X(2) ^ 5# – X(3) ^ 5# – X(4) ^ 5#
322 PD1 = -ABS(N(7))
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 8
1455 A(KLX) = X(KLX)

1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 IF M < -50 THEN 1999

1904 PRINT A(1), A(2), A(3), A(4)
1905 PRINT A(5), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [12, 15]. Copied by hand from the screen, the computer program’s complete output through JJJJ=32000 is shown below:

86 82 69 30
100 -43 -30476.5

69 82 86 30
100 -43 -19935.5

69 82 30 86
100 -43 -15466

133 110 27 84
144 0 -4576.5

84 27 110 133
144 0 16958

133 84 110 27
144 0 27217

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

At JJJJ=-4576.5, JJJJ=16958, and JJJJ=27217, M=0, which is optimal. See Lander and Parkin [5, 6].

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[5] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler’s Conjecture on Sums of Like Powers. The Bulletin of American Mathematical Society, Vol. 72, 1966, page 1079.

[6] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler’s Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[7] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[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] 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.

[10] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[11] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[12] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview

[13] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[14] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

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

[16] Jsun Yui Wong (2014, February 8), Testing the Nonlinear Integer Programming Solver with Lander and Parkin’s Counterexample to Euler’s Conjecture on Sums of Like Powers. https://computerprogramsandresults.wordpress.com/2014/02/08/testing

September 21, 2015 by 123writereadmath

A Large Nonlinear Integer Programming Problem of 32765 General Integer Variables Solved with QB64

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.5 but with 32765 unknowns instead of their 100 unknowns [4, pp. 416-417]. Specifically, the test example here is as follows:

Minimize

32765 32765
SIGMA X(i)^4 + [ SIGMA X(i) ]^2
i=1 i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 32765.

The following computer program uses qb64v1000-win [8, 9].

0 DEFINT J, K, B, X, A
2 DIM A(32765), X(32765)

81 FOR JJJJ = -32000 TO 32000
85 LB = -FIX(RND * 6)
86 UB = FIX(RND * 6)

87 br=-5+FIX(RND*11)

89 RANDOMIZE JJJJ
90 M = -1.5D+38
111 FOR J44 = 1 TO 32765

114 A(J44) = br
117 NEXT J44
128 FOR I = 1 TO 10000

129 FOR KQ = 1 TO 32765

130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32765)

167 IF RND < .5 THEN X(B) = (A(B) – 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32765
171 IF X(J44) < LB THEN X(J44) = LB
172 IF X(J44) > UB THEN X(J44) = UB
173 NEXT J44
482 SY = 0

483 FOR J44 = 1 TO 32765
485 SY = SY + X(J44) ^ 4
487 NEXT J44
488 SZ = 0
489 FOR J44 = 1 TO 32765
490 SZ = SZ + X(J44)
491 NEXT J44
492 PD1 = -SY – SZ ^ 2
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 32765

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 GOTO 128
1670 NEXT I

1781 PRINT A(1), A(2), A(3), A(222), A(32765), M, JJJJ, br

1788 rem print A(1111), A(11111), A(32762), A(32763), A(32764)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [8, 9]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31988 is shown below:

0 1 0 1 -1
-18128 -32000 1

0 1 1 -1 0
-19386 -31999 -3

0 0 0 0 0
-2 -31998 0

-1 -1 -1 1 0
-20820 -31997 2

1 -1 0 1 1
-21820 -31996 2

0 -1 -1 0 1
-17606 -31995 -4

0 0 0 0 0
0 -31994 -4

-1 -1 1 0 1
-18560 -31993 2

0 1 -1 -1 0
-17512 -31992 3

1 -1 -1 1 -1
-17814 -31991 -2

0 0 0 0 0
-2 -31990 0

0 0 0 0 0
0 -31989 -1

1 1 1 0 1
-18752 -31988 4

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

At JJJJ=-31994 and JJJJ=-31989, M=0, which is optimal. See Li and Sun [4, pp. 414-415].

Of the 32765 A’s, only the five A’s of line 1781 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 [8, 9], the wall-clock time for obtaining the output through JJJJ=-31988 was eight hours.

Acknowledgment

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

References

[1] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

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

[3] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey–Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[4] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[5] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[6] J. Plummer, L. S. Lasdon, M. Ahmed, Solving a Large Nonlinear Progammming Problem on a Vector Processing Computer, Annals of Operatons Research, Volume 14 (1988), Issue 1, pp.. 291-304.

[7] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[8] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview

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

September 6, 2015 by 123writereadmath

Solving with QB64 a Large Nonlinear Integer Programming Problem of 32765 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve Li and Sun’s Problem 14.5 but with 32765 unknowns instead of their 100 unknowns [12, pp. 416-417]. Specifically, the test example here is as follows:

Minimize

32765 32765
SIGMA X(i)^4 + [ SIGMA X(i) ]^2

i=1 i=1

subject to

-5 <= X(i) <= 5, X(i) integer, i=1, 2, 3,…, 32765.

One notes the starting solution vectors of line 111 through line 117, which are as follows:

111 FOR J44=1 TO 32765
114 A(J44)=-5+FIX(RND*11)
117 NEXT J44.

The following computer program uses qb64v1000-win [19, 20].

0 DEFINT J, K, B, X, A
2 DIM A(32765), X(32765)

81 FOR JJJJ = -32000 TO 32000
85 LB = -FIX(RND * 6)
86 UB = FIX(RND * 6)
89 RANDOMIZE JJJJ
90 M = -1.5D+38
111 FOR J44 = 1 TO 32765

114 A(J44) = -5 + FIX(RND * 11)
117 NEXT J44
128 FOR I = 1 TO 125000

129 FOR KQ = 1 TO 32765
130 X(KQ) = A(KQ)
131 NEXT KQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 32765)
167 IF RND < .5 THEN X(B) = (A(B) – 1) ELSE X(B) = (A(B) + 1)
169 NEXT IPP
170 FOR J44 = 1 TO 32765
171 IF X(J44) < LB THEN X(J44) = LB
172 IF X(J44) > UB THEN X(J44) = UB
173 NEXT J44
482 SY = 0

1781 PRINT A(1), A(2), A(3), A(222), A(32765), M, JJJJ

1788 PRINT A(1111), A(11111), A(32762), A(32763), A(32764)

1999 NEXT JJJJ

Modelled after the computer program in Wong [23], this BASIC computer program was run with qb64v1000-win [19, 20]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31993 is shown below:

-1 1 -1 0 -1
-17742 -32000
0 0 0 1 0

1 -1 1 0 1
-19202 -31999
0 0 0 -1 -1

-1 -1 0 0 0
-17074 -31998
0 1 0 1 0

0 0 0 0 0
-16120 -31997
1 1 0 -1 1

0 0 0 0 0
0 -31996
0 0 0 0 0

1 1 -1 1 1
-19790 -31995
0 0 1 1 0

1 1 1 1 -1
-29824 -31994
1 1 -1 1 -1

0 0 0 0 0
-16178 -31993
-1 0 0 -1 0

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

At JJJJ=-31996, M=0, which is optimal. See Li and Sun [12, p. 416].

Of the 32765 A’s, only the 10 A’s of line 1781 and line 1788 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 [19, 20], the wall-clock time for obtaining the output through JJJJ=-31993 was eleven hours.

For a computer program involving a mix of continuous variables and integer variables, see Wong [22], for instance.

Acknowledgment

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

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. – Oct., 1967), pp. 915-957.

[3] F. Cajori (1911) Historical Note on the Newton-Raphson Method of Approximation. The American Mathematical Monthly, Volume 18 #2, pp. 29-32.

[4] George B. Dantzig, Discrete-Variable Extrenum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[5] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, 36:307-339, 1986.

[6] D. M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

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

[8] M. Junger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Woolsey–Editors, 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer, 2010 Edition. eBook; ISBN 978-3-540-68279-0

[9] Jack Lashover (November 12, 2012). Monte Carlo Marching. http://www.academia.edu/5481312/MONTE_ CARLO_MARCHING

[10] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. – Dec., 1966), pp. 1098-1112.

[11] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May – June, 1967), p. 578.

[12] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer Science+Business Media,LLC (2006). http://www.books.google.ca/books?isbn=0387329951

[13] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[14] J. Plummer, L. S. Lasdon, M. Ahmed, Solving a Large Nonlinear Progammming Problem on a Vector Processing Computer, Annals of Operations Research, Volume 14 (1988), Issue 1, pp.. 291-304.

[15] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[16] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Elsevier Science Ltd (1989).

[17] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

[18] S. Surjanovic, Zakharov Function. http://www.sfu.ca/~ssurjano/zakharov.html

[19] E.K. Virtanen (2008-05-26). “Interview With Galleon”,
http://www.basicprogramming.org/pcopy/issue70/#galleoninterview

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

[21] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[22] Jsun Yui Wong (2013, July 16). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables, Sixth Edition, http://myblogsubstance.typepad.com/substance/2013/07/

[23] Jsun Yui Wong (2015, February 11). General Mixed Integer Nonlinear Programming (MINLP) Solver Using Cold Starts To Solve Li and Sun’s Problem 14.5 but Involving 15120 General Integer Variables instead of Their 100 General Integer Variables. http://myblogsubstance.typepad.com/substance/2015/02/