Auxiliary Problem Principle and Proximal Point Methods

Abstract

An extension of the auxiliary problem principle to variational inequalities with non-symmetric multi-valued operators in Hilbert spaces is studied. This extension concerns the case that the operator is split into the sum of a single-valued operator \cal F, possessing a kind of pseudo Dunn property, and a maximal monotone operator \cal L. The current auxiliary problem is k constructed by fixing \cal F at the previous iterate, whereas \cal L (or its single-valued approximation {\cal L}k) k is considered at a variable point. Using auxiliary operators of the form {\cal L}k+χ\nabla h, with χk>0, the standard for the auxiliary problem principle assumption of the strong convexity of the function h can be weakened exploiting mutual properties of \cal L and h. Convergence of the general scheme is analyzed and some applications are sketched briefly.