Sequential and parallel projection algorithms for feasibility and optimization

Abstract

The convex feasibility problem of finding a point in the intersection of finitely many nonempty closed convex sets in the Euclidean space has many applications in various fields of science and technology, particularly in problems of image reconstruction from projections, in solving the fully discretized inverse problem in radiation therapy treatment planning, and in other image processing problems. Solving systems of linear equalities and/or inequalities is one of them. Many of the existing algorithms use projections onto the sets and may: (i) employ orthogonal-, entropy-, or other Bregman-projections, (ii) be structurally sequential, parallel, block-iterative, or of the string-averaging type, (iii) asymptotically converge when the underlying system is, or is not, consistent, (iv) solve the convex feasibility problem or find the projection of a given point onto the intersection of the convex sets, (v) have good initial behavior patterns when some of their parameters are appropriately chosen.