Extended auxiliary problem principle to variational inequalities involving multi-valued operators
Alexander Kaplan, Rainer Tichatschke · Optimization · 2004
An extension of the auxiliary problem principle to variational inequalities with non-symmetric multi-valued operators in Hilbert spaces is studied. This extension supposes that the operator of the variational inequality is split up into the sum of a maximal monotone operator and a single-valued operator , which is linked with a sequence of non-symmetric components of auxiliary operators by a kind of pseudo Dunn property. The current auxiliary problem is constructed by fixing at the previous iterate, whereas is considered at a variable point. Using auxiliary operators of the form , the standard assumption of the strong convexity of the function h is weakened by exploiting mutual properties of and h. Convergence of the general scheme is analysed allowing that the auxiliary problems are solved approximately. Some applications are sketched briefly.