Auxiliary problem principle extended to variational inequalities
Journal of Optimization Theory and Applications
Mathematical Programming: Series A and B
Family of perturbation methods for variational inequalities
Journal of Optimization Theory and Applications
Progressive regularization of variational inequalities and decomposition algorithms
Journal of Optimization Theory and Applications
Approximate iterations in Bregman-function-based proximal algorithms
Mathematical Programming: Series A and B
Journal of Optimization Theory and Applications
Mathematics of Operations Research
SIAM Journal on Optimization
Relaxed proximal point algorithms for variational inequalities with multi-valued operators
Optimization Methods & Software
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An extension of the auxiliary problem principle for solving variational inequalities with maximal monotone operators is studied. The main idea of this approach consists in an application of Bregman functions for constructing symmetric (regularizing) components of the auxiliary operators. This provides an interior point effect, i.e., auxiliary problems can be treated as unconstrained ones. However, up to now, such Bregman functions were proposed only for linearly constrained variational inequalities or in the case where the constraint set K is a ball. In this article, considering a slightly modified concept of a Bregman function, we introduce appropriate functions for a wide class of sets K including, in particular, convex sets described by a system of inequalities with affine and strictly convex functions. The convergence analysis allows that the auxiliary problems are solved inexactly by using a sort of error summability criterion.