An inexact alternating direction method for solving a class of structured variational inequalities

The alternating direction method is mainly adopted to solve large-scale variational inequality problems with separable structure. The method is effective because it solves the original high-dimensional variational inequality problem by solving a series of much easier low-dimensional subproblems. In this paper, we present an inexact alternating directions method. Compared with the quadratic proximal alternating direction methods, the proposed method solves a series of related systems of nonlinear equations instead of a series of sub-VIs. The inexact criteria are more relaxed than the ones used by He et al. [7]. The generated sequence is Fejer monotone with respect to the solution set and the convergence is proved under suitable conditions.

[1]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[2]  Min Li,et al.  An LQP-Based Decomposition Method for Solving a Class of Variational Inequalities , 2011, SIAM J. Optim..

[3]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

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

[5]  Abdellah Bnouhachem,et al.  A projection-based prediction-correction method for structured monotone variational inequalities , 2008, Appl. Math. Comput..

[6]  B. Martinet.  Brève communication. Régularisation d'inéquations variationnelles par approximations successives , 1970 .

[7]  Xiaoming Yuan,et al.  On the O(1/t) Convergence Rate of Alternating Direction Method with Logarithmic-Quadratic Proximal Regularization , 2012, SIAM J. Optim..

[8]  Robert R. Meyer,et al.  A variable-penalty alternating directions method for convex optimization , 1998, Math. Program..

[9]  R. Tyrrell Rockafellar,et al.  Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming , 1976, Math. Oper. Res..

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

[11]  R. Glowinski.  Lectures on Numerical Methods for Non-Linear Variational Problems , 1981 .

[12]  Bingsheng He,et al.  A new inexact alternating directions method for monotone variational inequalities , 2002, Math. Program..

[13]  Bingsheng He,et al.  Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities , 1998, Oper. Res. Lett..

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

[15]  Marc Teboulle,et al.  A proximal-based decomposition method for convex minimization problems , 1994, Math. Program..