Strong convergence of an inexact projected subgradient method for mixed variational inequalities

In this article, we propose an easily implementable algorithm method for solving mixed variational inequalities. The proposed method combines two strategies: inexact projected subgradient methods and regularization techniques. We prove that the sequence generated by the method is strongly convergent to a solution of the problem in Hilbert spaces, whenever it exists. The results presented in this article generalize and improve some recent results.

[1]  W. Han,et al.  On the finite element method for mixed variational inequalities arising in elastoplasticity , 1995 .

[2]  Alfredo N. Iusem,et al.  On the projected subgradient method for nonsmooth convex optimization in a Hilbert space , 1998, Math. Program..

[3]  I. Konnov.  Combined Relaxation Methods for Variational Inequalities , 2000 .

[4]  Alfredo N. Iusem,et al.  Extension of Subgradient Techniques for Nonsmooth Optimization in Banach Spaces , 2001 .

[5]  Descent methods for mixed variational inequalities in a Hilbert space , 2001 .

[6]  Hai Yang,et al.  Inexact Implicit Method with Variable Parameter for Mixed Monotone Variational Inequalities , 2001 .

[7]  Igor V. Konnov,et al.  A Combined Relaxation Method for a Class of Nonlinear Variational Inequalities , 2002 .

[8]  I. V. Konnov,et al.  Mixed variational inequalities and economic equilibrium problems , 2002 .

[9]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[10]  P. Pardalos,et al.  Equilibrium problems : nonsmooth optimization and variational inequality models , 2004 .

[11]  Jean-Jacques Strodiot,et al.  A Bundle Method for Solving Variational Inequalities , 2004, SIAM J. Optim..

[12]  L. Muu,et al.  On the Contraction and Nonexpansiveness Properties of the Marginal Mappings in Generalized Variational Inequalities Involving Co-Coercive Operators , 2005 .

[13]  Lu-Chuan Zeng,et al.  Convergence analysis of a modified inexact implicit method for general mixed monotone variational inequalities , 2005, Math. Methods Oper. Res..

[14]  Abdellah Bnouhachem.  A self-adaptive method for solving general mixed variational inequalities , 2005 .

[15]  P. Maingé.  Strong Convergence of Projected Subgradient Methods for Nonsmooth and Nonstrictly Convex Minimization , 2008 .

[16]  Nan-Jing Huang,et al.  A projected subgradient method for solving generalized mixed variational inequalities , 2008, Oper. Res. Lett..

[17]  L. D. Muu,et al.  Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model , 2009, J. Optimization Theory and Applications.

[18]  Nan-jing Huang,et al.  The generalized f-projection operator and set-valued variational inequalities in Banach spaces☆ , 2009 .

[19]  Jen-Chih Yao,et al.  Existence of vector mixed variational inequalities in Banach spaces , 2009 .

[20]  Zhaohan Sheng,et al.  A modified inexact implicit method for mixed variational inequalities , 2010, J. Comput. Appl. Math..

[21]  Jen-Chih Yao,et al.  Well-posedness by perturbations of mixed variational inequalities in Banach spaces , 2010, Eur. J. Oper. Res..

[22]  Le Dung Muu,et al.  One step from DC optimization to DC mixed variational inequalities , 2010 .

[23]  Jen-Chih Yao,et al.  An iterative algorithm for system of mixed variational-like inequalities , 2010 .

[24]  Paul-Emile Maingé,et al.  Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints , 2010, Eur. J. Oper. Res..

[25]  Nan-Jing Huang,et al.  Stability Analysis for Minty Mixed Variational Inequality in Reflexive Banach Spaces , 2010, J. Optim. Theory Appl..

[26]  Susana Scheimberg,et al.  An inexact subgradient algorithm for Equilibrium Problems , 2011 .