American Institute of Mathematical Sciences

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April  2015, 11(2): 345-364. doi: 10.3934/jimo.2015.11.345

A new auxiliary function method for systems of nonlinear equations

Zhiyou Wu 1, , Fusheng Bai 1, , Guoquan Li 1, and  Yongjian Yang 2,

1. 

School of Mathematics, Chongqing Normal University, Chongqing 401331, China, China, China
2. 

Department of Mathematics, Shanghai University, Shanghai 200444

Received  November 2013 Revised  May 2014 Published  September 2014

In this paper, we present a new global optimization method to solve nonlinear systems of equations. We reformulate given system of nonlinear equations as a global optimization problem and then give a new auxiliary function method to solve the reformulated global optimization problem. The new auxiliary function proposed in this paper can be a filled function, a quasi-filled function or a strict filled function with appropriately chosen parameters. Several numerical examples are presented to illustrate the efficiency of the present approach.
Keywords: Auxiliary function method, global optimization problems., nonlinear equations.
Mathematics Subject Classification: 90C2.
Citation: Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial and Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345
References:
[1]

S. C. Billups and L. T. Watson, A probability-one homotopy algorithm for nonsmooth equations and mixed complementarity problems, SIAM Journal on Optimization, 12 (2002), 606-626. doi: 10.1137/S105262340037758X.

[2]

X. Chen, L. Qi and Y. F. Yang, Lagrangian globalization methods for nonlinear complementarity problem, Journal of Optimization Theory and Applications, 112 (2002), 77-95. doi: 10.1023/A:1013092412197.

[3]

B. Cetin, J. Barhen and J. Burdick, Terminal repeller unconstrained subenergy tunneling (TRUST) for fast global optimization, J. Optim. Theory Appl., 77 (1993), 97-126. doi: 10.1007/BF00940781.

[4]

A. R. Conn, N. I. M. Gould and P. L. Toint, Trust Region Methods, SIAM, Philadelphia, USA, 2000. doi: 10.1137/1.9780898719857.

[5]

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, USA, 1996. doi: 10.1137/1.9781611971200.

[6]

C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer and C. A. Schweiger, Handbook of Test Problems in Local and Global Optimization, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1999. doi: 10.1007/978-1-4757-3040-1.

[7]

R. Ge, A filled function method for finding a global minimizer of a function of several variables, Mathematical Programming, 46 (1990), 191-204. doi: 10.1007/BF01585737.

[8]

R. P. Ge and Y. Qin, A class of filled functions for finding global minimizers of a function of several variables, Journal of Optimization Theory and Applications, 54 (1987), 241-252. doi: 10.1007/BF00939433.

[9]

C. Kanzow, Global optimization techniques for mixed complementarity problems, Journal of Global Optimization, 16 (2000), 1-21. doi: 10.1023/A:1008331803982.

[10]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Phildelphia, PA, 1995. doi: 10.1137/1.9781611970944.

[11]

J. Kostrowicki and L. Piela, Diffusion equation method of global minimization: Performance for standard test functions, J. Optim. Theory Appl., 69 (1991), 269-284. doi: 10.1007/BF00940643.

[12]

X. Liu, A computable filled function used for global optimization, Appllied Mathematica and Computation, 126 (2002), 271-278. doi: 10.1016/S0096-3003(00)00157-0.

[13]

X. Liu, A new filled function applied to global optimization, Computers and Operations Research, 31 (2004), 61-80. doi: 10.1016/S0305-0548(02)00154-5.

[14]

J. More, G. Burton and K. Hillstrom, User guide for MINPACK-1, Argonne National Labs Report ANL-80-74, Argonne, Illinois, 1980.

[15]

J. L. Nazareth and L. Qi, Globalization of Newton's methods for solving nonlinear equations, Numerical linear algebra with applications, 3 (1996), 239-249.

[16]

H. Sellami and S. M. Robinson, Implementation of a continuation method for normal maps, Mathematical Programming, 76 (1997), 563-578. doi: 10.1007/BF02614398.

[17]

X. J. Tong, L. Qi and Y. F. Yang, The Lagrangian globalization method for nonsmooth constrained equations, Computational Optimization and Applications, 33 (2006), 89-109. doi: 10.1007/s10589-005-5960-9.

[18]

Z. Y. Wu, M. Mammadov, F. S. Bai and Y. J. Yang, A filled function method for nonlinear equations, Applied Mathematics and Computation, 189 (2007), 1196-1204. doi: 10.1016/j.amc.2006.11.183.

[19]

Z. Xu, H. X. Huang, P. M. Pardalos and C. X. Xu, Filled functions for unconstrained global optimization, Journal of Global Optimization, 20 (2001), 49-65. doi: 10.1023/A:1011207512894.

[20]

W. X. Zhu, Globally concavizied filled function method for the box constrained global minimization problem, Optimization Methods and Software, 21 (2006), 653-666. doi: 10.1080/10556780600628188.

show all references

References:
[1]

S. C. Billups and L. T. Watson, A probability-one homotopy algorithm for nonsmooth equations and mixed complementarity problems, SIAM Journal on Optimization, 12 (2002), 606-626. doi: 10.1137/S105262340037758X.

[2]

X. Chen, L. Qi and Y. F. Yang, Lagrangian globalization methods for nonlinear complementarity problem, Journal of Optimization Theory and Applications, 112 (2002), 77-95. doi: 10.1023/A:1013092412197.

[3]

B. Cetin, J. Barhen and J. Burdick, Terminal repeller unconstrained subenergy tunneling (TRUST) for fast global optimization, J. Optim. Theory Appl., 77 (1993), 97-126. doi: 10.1007/BF00940781.

[4]

A. R. Conn, N. I. M. Gould and P. L. Toint, Trust Region Methods, SIAM, Philadelphia, USA, 2000. doi: 10.1137/1.9780898719857.

[5]

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, USA, 1996. doi: 10.1137/1.9781611971200.

[6]

C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer and C. A. Schweiger, Handbook of Test Problems in Local and Global Optimization, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1999. doi: 10.1007/978-1-4757-3040-1.

[7]

R. Ge, A filled function method for finding a global minimizer of a function of several variables, Mathematical Programming, 46 (1990), 191-204. doi: 10.1007/BF01585737.

[8]

R. P. Ge and Y. Qin, A class of filled functions for finding global minimizers of a function of several variables, Journal of Optimization Theory and Applications, 54 (1987), 241-252. doi: 10.1007/BF00939433.

[9]

C. Kanzow, Global optimization techniques for mixed complementarity problems, Journal of Global Optimization, 16 (2000), 1-21. doi: 10.1023/A:1008331803982.

[10]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Phildelphia, PA, 1995. doi: 10.1137/1.9781611970944.

[11]

J. Kostrowicki and L. Piela, Diffusion equation method of global minimization: Performance for standard test functions, J. Optim. Theory Appl., 69 (1991), 269-284. doi: 10.1007/BF00940643.

[12]

X. Liu, A computable filled function used for global optimization, Appllied Mathematica and Computation, 126 (2002), 271-278. doi: 10.1016/S0096-3003(00)00157-0.

[13]

X. Liu, A new filled function applied to global optimization, Computers and Operations Research, 31 (2004), 61-80. doi: 10.1016/S0305-0548(02)00154-5.

[14]

J. More, G. Burton and K. Hillstrom, User guide for MINPACK-1, Argonne National Labs Report ANL-80-74, Argonne, Illinois, 1980.

[15]

J. L. Nazareth and L. Qi, Globalization of Newton's methods for solving nonlinear equations, Numerical linear algebra with applications, 3 (1996), 239-249.

[16]

H. Sellami and S. M. Robinson, Implementation of a continuation method for normal maps, Mathematical Programming, 76 (1997), 563-578. doi: 10.1007/BF02614398.

[17]

X. J. Tong, L. Qi and Y. F. Yang, The Lagrangian globalization method for nonsmooth constrained equations, Computational Optimization and Applications, 33 (2006), 89-109. doi: 10.1007/s10589-005-5960-9.

[18]

Z. Y. Wu, M. Mammadov, F. S. Bai and Y. J. Yang, A filled function method for nonlinear equations, Applied Mathematics and Computation, 189 (2007), 1196-1204. doi: 10.1016/j.amc.2006.11.183.

[19]

Z. Xu, H. X. Huang, P. M. Pardalos and C. X. Xu, Filled functions for unconstrained global optimization, Journal of Global Optimization, 20 (2001), 49-65. doi: 10.1023/A:1011207512894.

[20]

W. X. Zhu, Globally concavizied filled function method for the box constrained global minimization problem, Optimization Methods and Software, 21 (2006), 653-666. doi: 10.1080/10556780600628188.

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