Auxiliary Problem Principle and Proximal Point Methods

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 splitted into the sum of a single-valued operator F , possessing a kind of pseudo Dunn property, and a maximal monotone operator Q. The current auxiliary problem is constructed by fixing F at the previous iterate, whereas Q (or its single-valued approximation Q) is considered at a variable point. Using auxiliary operators of the form L+χk∇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 Q and h. Convergence of the general scheme is analysed and some applications are sketched briefly.