A globally convergent Levenberg-Marquardt method for the least l2-norm solution of nonlinear inequalities

Abstract The least l 2 -norm solution for a possibly inconsistent system of nonlinear inequalities is studied in this paper. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a smoothing Levenberg–Marquardt method is applied to solve the parameterized optimization problems. The global convergence of the proposed method is established under suitable assumptions.

[1]  Defeng Sun,et al.  Improving the convergence of non-interior point algorithms for nonlinear complementarity problems , 2000, Math. Comput..

[2]  Pu-yan Nie,et al.  A filter method for solving nonlinear complementarity problems , 2005, Appl. Math. Comput..

[3]  Pu-yan Nie.  A Null Space Approach for Solving Nonlinear Complementarity Problems , 2006 .

[4]  Zheng-Hai Huang,et al.  Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem , 2003 .

[5]  Jinyan Fan,et al.  A derivative-free filter method for solving nonlinear complementarity problems , 2005, Appl. Math. Comput..

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

[7]  M. Sahba,et al.  On the solution of nonlinear inequalities in a finite number of iterations , 1985 .

[8]  Patrick T. Harker,et al.  A Noninterior Continuation Method for Quadratic and Linear Programming , 1993, SIAM J. Optim..

[9]  Y. Ye,et al.  On Homotopy-Smoothing Methods for Box-Constrained Variational Inequalities , 1999 .

[10]  Paul Tseng,et al.  Non-Interior continuation methods for solving semidefinite complementarity problems , 2003, Math. Program..

[11]  James V. Burke,et al.  The Global Linear Convergence of a Noninterior Path-Following Algorithm for Linear Complementarity Problems , 1998, Math. Oper. Res..

[12]  Olvi L. Mangasarian,et al.  A class of smoothing functions for nonlinear and mixed complementarity problems , 1996, Comput. Optim. Appl..

[13]  David Q. Mayne,et al.  Solving nonlinear inequalities in a finite number of iterations , 1981 .

[14]  Xiaojun Chen,et al.  Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities , 1998, Math. Comput..

[15]  L. Qi,et al.  Solving variational inequality problems via smoothing-nonsmooth reformulations , 2001 .

[16]  J. Z. Zhang,et al.  Nonmonotone Levenberg–Marquardt Algorithms and Their Convergence Analysis , 1997 .

[17]  Naiyang Deng,et al.  A Family of Scaled Factorized Broyden-Like Methods for Nonlinear Least Squares Problems , 1999, SIAM J. Optim..

[18]  J. Dennis,et al.  Convergence theory for the structured BFGS secant method with an application to nonlinear least squares , 1989 .

[19]  Defeng Sun,et al.  A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities , 2000, Math. Program..

[20]  Paul Tseng,et al.  Merit functions for semi-definite complemetarity problems , 1998, Math. Program..

[21]  Y. Ye,et al.  On Homotopy-Smoothing Methods for Variational Inequalities , 1999 .

[22]  H. Yabe,et al.  Factorized quasi-Newton methods for nonlinear least squares problems , 1991, Math. Program..

[23]  Christian Kanzow,et al.  Some Noninterior Continuation Methods for Linear Complementarity Problems , 1996, SIAM J. Matrix Anal. Appl..