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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 & 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.  doi: 10.1137/S105262340037758X.  Google Scholar

[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.  doi: 10.1023/A:1013092412197.  Google Scholar

[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.  doi: 10.1007/BF00940781.  Google Scholar

[4]

A. R. Conn, N. I. M. Gould and P. L. Toint, Trust Region Methods,, SIAM, (2000).  doi: 10.1137/1.9780898719857.  Google Scholar

[5]

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations,, SIAM, (1996).  doi: 10.1137/1.9781611971200.  Google Scholar

[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, (1999).  doi: 10.1007/978-1-4757-3040-1.  Google Scholar

[7]

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

[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.  doi: 10.1007/BF00939433.  Google Scholar

[9]

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

[10]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations,, SIAM, (1995).  doi: 10.1137/1.9781611970944.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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.  doi: 10.1007/s10589-005-5960-9.  Google Scholar

[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.  doi: 10.1016/j.amc.2006.11.183.  Google Scholar

[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.  doi: 10.1023/A:1011207512894.  Google Scholar

[20]

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

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.  doi: 10.1137/S105262340037758X.  Google Scholar

[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.  doi: 10.1023/A:1013092412197.  Google Scholar

[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.  doi: 10.1007/BF00940781.  Google Scholar

[4]

A. R. Conn, N. I. M. Gould and P. L. Toint, Trust Region Methods,, SIAM, (2000).  doi: 10.1137/1.9780898719857.  Google Scholar

[5]

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations,, SIAM, (1996).  doi: 10.1137/1.9781611971200.  Google Scholar

[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, (1999).  doi: 10.1007/978-1-4757-3040-1.  Google Scholar

[7]

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

[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.  doi: 10.1007/BF00939433.  Google Scholar

[9]

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

[10]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations,, SIAM, (1995).  doi: 10.1137/1.9781611970944.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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.  doi: 10.1007/s10589-005-5960-9.  Google Scholar

[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.  doi: 10.1016/j.amc.2006.11.183.  Google Scholar

[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.  doi: 10.1023/A:1011207512894.  Google Scholar

[20]

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

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