Event № 303

The convex feasibility problem (CFP) is defined as the problem of finding a point lying in the intersection of given closed and convex sets C_i, i=1,2,...,m, which are called constraints. A typical example of such a problem is solving a system of linear inequalities. In this talk I will discuss an algorithmic approach to solving the CFP reformulated as the common fixed point problem (CFPP) for quasi-nonexpansive mappings U_i, i=1,...,m, satisfying the relation Fix(U_i)=C_i. The solution strategy, roughly speaking, exploits the underlying assumption that the computation of each U_i is simple. The presented method, called Modular String Averaging, is based on three simple concepts: relaxation, convex combination and composition of the input operators U_i. The algorithm, despite its simple description, covers a wide class of methods commonly used for CFPs and CFPPs. It will be shown that under certain additional assumptions, a similar algorithm can be adapted to variational inequalities. This talk is partially based on my PhD thesis and on some recent results obtained jointly with Simeon Reich.