Extended auxiliary problem principle using Bregman distances

Abstract

An extension of the Auxiliary Problem Principle (cf., G. Cohen (1980). Auxiliary problem principle and decomposition of optimization problems. JOTA, 32, 277–305; G. Cohen (1988). Auxiliary problem principle extended to variational inequalities. JOTA, 59, 325–333.) for solving variational inequalities with maximal monotone operators is studied. Using Bregman functions to construct the symmetric components of the auxiliary operators, an “interior point effect” is provided, i.e. auxiliary problems can be treated as unconstrained ones.

 For the sake of brevity we avoid here a repetition of results and facts viewed in [14,17] and refer only to the investigations which are directly connected with the main content of this article.

In this general framework, classical and Bregman-function based proximal methods can be considered as particular cases.

The convergence analysis allows that the auxiliary problems are solved inexactly with a sort of error summability criterion.

Keywords:

• Variational inequalities
• Auxiliary problem principle
• Proximal-like regularization
• Bregman distances
• AMS Subject Classifications 2000: 47J20
• 47H05
• 65J20
• 65K10
• 90C25

Notes

 For the sake of brevity we avoid here a repetition of results and facts viewed in [14,17] and refer only to the investigations which are directly connected with the main content of this article.

 With K in place of

monotone operators approximating

by a type of Mosco convergence.

 That is, with a given m > 0,

holds for all

.

 In this case we take

in the EPAP-method, ∂_ε denotes the ε-subdifferential. Because

, the relation (2) is evident.