An inexact interior point proximal method for the variational inequality problem

We propose an infeasible interior proximal method for solving variational inequality problems with maximal monotone operators and linear constraints. The interior proximal method proposed by Auslender, Teboulle and Ben-Tiba [3] is a proximal method using a distance-like barrier function and it has a global convergence property under mild assumptions. However, this method is applicable only to problems whose feasible region has nonempty interior. The algorithm we propose is applicable to problems whose feasible region may have empty interior. Moreover, a new kind of inexact scheme is used. We present a full convergence analysis for our algorithm.

[1]  Xingshi He,et al.  Introduction to Optimization , 2015, Applied Evolutionary Algorithms for Engineers Using Python.

[2]  Masao Fukushima,et al.  "Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods" , 2010 .

[3]  M. Teboulle,et al.  Asymptotic cones and functions in optimization and variational inequalities , 2002 .

[4]  Regina Sandra Burachik,et al.  A Relative Error Tolerance for a Family of Generalized Proximal Point Methods , 2001, Math. Oper. Res..

[5]  Benar Fux Svaiter,et al.  An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions , 2000, Math. Oper. Res..

[6]  Marc Teboulle,et al.  Interior Proximal and Multiplier Methods Based on Second Order Homogeneous Kernels , 1999, Math. Oper. Res..

[7]  Jonathan Eckstein,et al.  Approximate iterations in Bregman-function-based proximal algorithms , 1998, Math. Program..

[8]  Alfredo N. Iusem,et al.  An interior point method with Bregman functions for the variational inequality problem with paramonotone operators , 1998, Math. Program..

[9]  K. Kiwiel.  Proximal Minimization Methods with Generalized Bregman Functions , 1997 .

[10]  Marc Teboulle,et al.  Convergence of Proximal-Like Algorithms , 1997, SIAM J. Optim..

[11]  Mounir Haddou,et al.  An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities , 1995, Math. Program..

[12]  Marc Teboulle,et al.  Entropy-Like Proximal Methods in Convex Programming , 1994, Math. Oper. Res..

[13]  Paul Tseng,et al.  On the convergence of the exponential multiplier method for convex programming , 1993, Math. Program..

[14]  Marc Teboulle,et al.  Entropic Proximal Mappings with Applications to Nonlinear Programming , 1992, Math. Oper. Res..

[15]  P. P. B. Eggermont,et al.  Multiplicative iterative algorithms for convex programming , 1990 .

[16]  D. Pascali,et al.  Nonlinear mappings of monotone type , 1979 .

[17]  R. Rockafellar,et al.  On the subdifferentiability of convex functions , 1965 .

[18]  A. Hoffman.  On approximate solutions of systems of linear inequalities , 1952 .

[19]  A. Iusem,et al.  Set-valued mappings and enlargements of monotone operators , 2008 .

[20]  M. Fukushima,et al.  An Infeasible Interior Proximal Method for Convex Programming Problems with Linear Constraints , 2005 .

[21]  Marc Teboulle,et al.  A Logarithmic-Quadratic Proximal Method for Variational Inequalities , 1999, Comput. Optim. Appl..

[22]  A. Iusem.  On some properties of paramonotone operators. , 1998 .

[23]  B. Lemaire.  On the Convergence of Some Iterative Methods for Convex Minimization , 1995 .

[24]  E. Chong,et al.  Introduction to optimization , 1987 .

[25]  F. Browder.  Nonlinear operators and nonlinear equations of evolution in Banach spaces , 1976 .

[26]  R. Rockafellar.  On the maximality of sums of nonlinear monotone operators , 1970 .