Spectral gradient projection method for monotone nonlinear equations with convex constraints

A spectral gradient projection algorithm for monotone nonlinear equations with convex constraints is proposed. This new procedure is obtained by combining a modified spectral gradient method and a projection method. A relevant property of the presented method is that the computation of the iteration sequence does not require the solution of any subproblem. The algorithm is shown to be globally convergent under mild assumptions and it can be also applied to nonsmooth equations. Preliminary numerical tests show that the proposed method works quite well even for large scale problems.

[1]  James A. Momoh,et al.  Optimal power flow : solution techniques, requirements, and challenges , 1996 .

[2]  Duan Li,et al.  Monotonicity of Fixed Point and Normal Mappings Associated with Variational Inequality and Its Application , 2000, SIAM J. Optim..

[3]  José Mario Martínez,et al.  Nonmonotone Spectral Projected Gradient Methods on Convex Sets , 1999, SIAM J. Optim..

[4]  J. Pang,et al.  A Trust Region Method for Constrained Nonsmooth Equations , 1994 .

[5]  A. Morgan,et al.  A methodology for solving chemical equilibrium systems , 1987 .

[6]  José Mario Martínez,et al.  Spectral residual method without gradient information for solving large-scale nonlinear systems of equations , 2006, Math. Comput..

[7]  C. Kanzow.  Levenberg-Marquardt methods for constrained nonlinear equations with strong local convergence properties , 2004 .

[8]  Weijun Zhou,et al.  Spectral gradient projection method for solving nonlinear monotone equations , 2006 .

[9]  Layne T. Watson,et al.  A Probability-One Homotopy Algorithm for Nonsmooth Equations and Mixed Complementarity Problems , 2002, SIAM J. Optim..

[10]  L. Grippo,et al.  A nonmonotone line search technique for Newton's method , 1986 .

[11]  Philip G. Hill,et al.  Power generation , 1927, Journal of the A.I.E.E..

[12]  Mikhail V. Solodov,et al.  A Globally Convergent Inexact Newton Method for Systems of Monotone Equations , 1998 .

[13]  Zhensheng Yu.  Solving bound constrained optimization via a new nonmonotone spectral projected gradient method , 2008 .

[14]  W. Hager,et al.  Large Scale Optimization : State of the Art , 1993 .

[15]  Chuanwei Wang,et al.  A projection method for a system of nonlinear monotone equations with convex constraints , 2007, Math. Methods Oper. Res..

[16]  Defeng Sun,et al.  A feasible semismooth asymptotically Newton method for mixed complementarity problems , 2002, Math. Program..

[17]  M. Fukushima,et al.  Erratum to Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints , 2005 .

[18]  Stefania Bellavia,et al.  An affine scaling trust-region approach to bound-constrained nonlinear systems , 2003 .

[19]  Panos M. Pardalos,et al.  Encyclopedia of Optimization , 2006 .

[20]  Xiaojiao Tong,et al.  A smoothing projected Newton-type method for semismooth equations with bound constraints , 2005 .

[21]  Alexander I. Barvinok.  Computing the Ehrhart quasi-polynomial of a rational simplex , 2006, Math. Comput..

[22]  Alexander P. Morgan,et al.  Chemical equilibrium systems as numerical test problems , 1990, TOMS.

[23]  Michael Ulbrich,et al.  Nonmonotone Trust-Region Methods for Bound-Constrained Semismooth Equations with Applications to Nonlinear Mixed Complementarity Problems , 2000, SIAM J. Optim..

[24]  J. Borwein,et al.  Two-Point Step Size Gradient Methods , 1988 .

[25]  S. M. Robinson,et al.  Homotopies Based on Nonsmooth Equations for Solving Nonlinear Variational Inequalities , 1996 .

[26]  Liqun Qi,et al.  Active-Set Projected Trust-Region Algorithm for Box-Constrained Nonsmooth Equations , 2004 .

[27]  M. V. Solodovy,et al.  Newton-type Methods with Generalized Distances For Constrained Optimization , 1997 .

[28]  Christodoulos A. Floudas,et al.  Finding all solutions of nonlinearly constrained systems of equations , 1995, J. Glob. Optim..

[29]  Liqun Qi,et al.  On the Convergence of a Trust-Region Method for Solving Constrained Nonlinear Equations with Degenerate Solutions , 2004 .

[30]  M. Raydan,et al.  Spectral Projected Gradient Methods , 2007 .

[31]  S. Dirkse,et al.  Mcplib: a collection of nonlinear mixed complementarity problems , 1995 .

[32]  Marcos Raydan,et al.  Nonmonotone Spectral Methods for Large-Scale Nonlinear Systems , 2003, Optim. Methods Softw..

[33]  Marcos Raydan,et al.  The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem , 1997, SIAM J. Optim..

[34]  Liqun Qi,et al.  The Lagrangian Globalization Method for Nonsmooth Constrained Equations , 2006, Comput. Optim. Appl..