# American Institute of Mathematical Sciences

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

## A new auxiliary function method for systems of nonlinear equations

 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.
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.

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##### 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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