Hilbertian Convex Feasibility Problem: Convergence of Projection Methods

Finding best approximation pairs relative to two closed convex sets in Hilbert spaces

“…which proves (13). Assume for the remainder of the proof that b n À a n -v or, equivalently, jjb n À a n jj-jjvjj: Since ð8nANÞ jjb n À a n jjX maxfjjb n À P A b n jj; jjP B a n À a n jjg X minfjjb n À P A b n jj; jjP B a n À a n jjg X jjvjj;…”

“…If A ¼ fag; (4) reduces to (3) and its solution is P B a: On the other hand, when the problem is consistent, i.e., A-Ba|; then (4) reduces to the well-known convex feasibility problem for two sets [4,13] and its solution set is fðx; xÞAX Â X :xAA-Bg: Formulation (4) captures a wide range of problems in applied mathematics and engineering [11,24,27,30,35]. The method of alternating projections applied to the sets A and B is perhaps the most straightforward algorithm to obtain a best approximation pair.…”

Mixed equilibrium problems and optimization problems

“…This kind of optimization problems has been studied extensively by many authors, see, for example, Bauschke and Borwein [18], Combettes [19], Deutsch and Yamada [20] and Xu [21] …”

Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach spaces

“…They prove that if ∞ n=1 ω n (i) = ∞ for each i ∈Ĩ , then their block-iterative algorithm converges to some point in the intersection of the sets C i , as long as all relaxation parameters λ n are confined to a closed interval of the form [ε, 2 − ε], ε > 0. Another block-iterative framework for solving the convex feasibility problem was proposed by Combettes [9,10] in general Hilbert spaces. In this scheme, as compared to the Aharoni-Censor scheme, the broader class of firmly nonexpansive mappings is involved in the iteration process, while a slightly more restrictive condition is imposed on the sequence of weights.…”