A filled function method for nonlinear systems of equalities and inequalities

In this paper a filled function method is suggested for solving nonlinear systems of equalities and inequalities. Firstly, the original problem is reformulated into an equivalent constrained global optimization problem. Subsequently, a new filled function with one parameter is constructed based on the special characteristics of the reformulated optimization problem. Some properties of the filled function are studied and discussed. Finally, an algorithm based on the proposed filled function for solving nonlinear systems of equalities and inequalities is presented. The objective function value can be reduced by half in each iteration of our filled function algorithm. The implementation of the algorithm on several test problems is reported with numerical results. Mathematical subject classification: 65K05, 90C30.

[1]  Boris Polyak Gradient methods for solving equations and inequalities , 1964 .

[2]  B. N. Pshenichnyi Newton's method for the solution of systems of equalities and inequalities , 1970 .

[3]  S. M. Robinson Extension of Newton's method to nonlinear functions with values in a cone , 1972 .

[4]  J. Daniel Newton's method for nonlinear inequalities , 1973 .

[5]  Ubaldo M. García-Palomares,et al.  A global quadratic algorithm for solving a system of mixed equalities and inequalities , 1981, Math. Program..

[6]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[7]  A. V. Levy,et al.  The Tunneling Algorithm for the Global Minimization of Functions , 1985 .

[8]  G. T. Timmer,et al.  Stochastic global optimization methods part I: Clustering methods , 1987, Math. Program..

[9]  C. Kelley Iterative Methods for Linear and Nonlinear Equations , 1987 .

[10]  R. Ge,et al.  A class of filled functions for finding global minimizers of a function of several variables , 1987 .

[11]  G. T. Timmer,et al.  Stochastic global optimization methods part II: Multi level methods , 1987, Math. Program..

[12]  Renpu Ge,et al.  A Filled Function Method for Finding a Global Minimizer of a Function of Several Variables , 1990, Math. Program..

[13]  Jacob Barhen,et al.  TRUST: A deterministic algorithm for global optimization , 1997 .

[14]  J. E. DENNIS,et al.  A Trust-Region Approach to Nonlinear Systems of Equalities and Inequalities , 1999, SIAM J. Optim..

[15]  I. Dikin Solution of Systems of Equalities and Inequalities by the Method of Interior Points , 2004 .

[16]  Zhi-You Wu,et al.  A filled function method for constrained global optimization , 2007, J. Glob. Optim..

[17]  Zhongping Wan,et al.  A filled function method for solving nonlinear complementarity problem , 2009 .

[18]  Yanping,et al.  Smoothing Newton-Like Method for the Solution of Nonlinear Systems of Equalities and Inequalities , 2009 .

[19]  Yongjian Yang,et al.  A new filled function method for nonlinear equations , 2009, Appl. Math. Comput..

[20]  Benedetta Morini,et al.  Trust-region quadratic methods for nonlinear systems of mixed equalities and inequalities , 2009 .

[21]  Yongjian Yang,et al.  Filled function method for nonlinear equations , 2010, J. Comput. Appl. Math..

[22]  Benedetta Morini,et al.  TRESNEI, a Matlab trust-region solver for systems of nonlinear equalities and inequalities , 2010, Computational Optimization and Applications.

[23]  Yongjian Yang,et al.  A new filled function method for constrained nonlinear equations , 2012, Appl. Math. Comput..