Iterative algorithms with seminorm‐induced oblique projections

Abstract

A definition of oblique projections onto closed convex sets that use seminorms induced by diagonal matrices which may have zeros on the diagonal is introduced. Existence and uniqueness of such projections are secured via directional affinity of the sets with respect to the diagonal matrices involved. A blockiterative algorithmic scheme for solving the convex feasibility problem, employing seminorm-induced oblique projections, is constructed and its convergence for the consistent case is established. The fully simultaneous algorithm converges also in the inconsistent case to the minimum of a certain proximity function.

Weighted Generalized Inverses, Oblique Projections, and Least-Squares Problems

“…For recent results on this subject the reader is referred to the papers by Sun and Wei [30], Stanimirović and Stanković [29], Wei and Wang [35], and Djordjević, Stanimirović, and Wei [11]. For applications to parallel computing, image processing, and many algorithmical results that use weighted generalized inverses with singular weights, the reader is referred to the papers by Censor, Gordon and Gordon [4], [5] and Censor and Elfving [2], [3]. The papers by Nashed and Votruba [24] and Nashed [23] and the books by Rao and Mitra [26] and Ben-Israel and Greville [1] are excellent references, which contain many results on weighted generalized inverses.…”

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. II. Singular weights

“…As follows from estimates (22), (23), (29), (33), (36), (38), (41) The paper [1] reviews the literature on direct and iterative methods of computing weighted pseudoinverses and weighted normal pseudosolutions. Noteworthy are the papers [31][32][33][34], which contain some algorithmic results that use weighted pseudoinversion with singular weights and consider problems of parallel computations for the algorithms proposed.…”