Some Remarks on the Convex Feasibility Problem and Best Approximation Problem

Abstract

In this paper we investigate several solution algorithms for the convex feasibility problem (CFP) and the best approximation problem (BAP) respectively. The algorithms analyzed are already known before, but by adequately reformulating the CFP or the BAP we naturally deduce the general projection method for the CFP from well-known steepest decent method for unconstrained optimization and we also give a natural strategy of updating weight parameters. In the linear case we show the connection of the two projection algorithms for the CFP and the BAP respectively. In addition, we establish the convergence of a method for the BAP under milder assumptions in the linear case. We also show by examples a Bauschke's conjecture is only partially correct.