Convergent Algorithm Based on Progressive Regularization for Solving Pseudomonotone Variational Inequalities

In this paper, we extend the Moreau-Yosida regularization of monotone variational inequalities to the case of weakly monotone and pseudomonotone operators. With these properties, the regularized operator satisfies the pseudo-Dunn property with respect to any solution of the variational inequality problem. As a consequence, the regularized version of the auxiliary problem algorithm converges. In this case, when the operator involved in the variational inequality problem is Lipschitz continuous (a property stronger than weak monotonicity) and pseudomonotone, we prove the convergence of the progressive regularization introduced in Refs. 1, 2.

[1]  L. Bregman.  The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[2]  Jong-Shi Pang,et al.  Iterative methods for variational and complementarity problems , 1982, Math. Program..

[3]  Guy Cohen,et al.  Progressive Regularization of Variational Inequalities and Decomposition Algorithms , 1998 .

[4]  G. Cohen.  Auxiliary problem principle and decomposition of optimization problems , 1980 .

[5]  N. El Farouq,et al.  Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method , 2001 .

[6]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[7]  S. Karamardian.  Complementarity problems over cones with monotone and pseudomonotone maps , 1976 .

[8]  Jean-Philippe Vial,et al.  Strong and Weak Convexity of Sets and Functions , 1983, Math. Oper. Res..

[9]  Jean-Pierre Crouzeix,et al.  Pseudomonotone variational inequality problems: Existence of solutions , 1997, Math. Program..

[10]  G. Cohen.  Auxiliary problem principle extended to variational inequalities , 1988 .

[11]  Patrice Marcotte,et al.  Co-Coercivity and Its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities , 1996, SIAM J. Optim..

[12]  J. Spingarn.  Submonotone mappings and the proximal point algorithm , 1982 .

[13]  Naïma El Farouq.  Algorithmes de résolution d'inéquations variationnelles , 1993 .

[14]  Jonathan Eckstein,et al.  Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming , 1993, Math. Oper. Res..

[15]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[16]  Jen-Chih Yao,et al.  Multi-valued variational inequalities with K-pseudomonotone operators , 1994 .

[17]  E. Zeidler.  Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators , 1989 .

[18]  G. Cohen.  Optimization by decomposition and coordination: A unified approach , 1978 .

[19]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..