Yair Censor We present a modification of Dykstra’s algorithm which allows us to avoid projections onto general convex sets. Instead, we calculate projections onto either a halfspace or onto the intersection of two halfspaces. Convergence of the algorithm is established and special choices of the halfspaces are proposed. The option to project onto halfspaces instead of general … Read more
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 block-iterative algorithmic scheme for solving the convex feasibility … Read more
Aperture-based inverse planning (ABIP) for intensity modulated radiation therapy (IMRT) treatment planning starts with external radiation fields (beams) that fully conform to the target(s) and then superimposes sub-fields called segments to achieve complex shaping of 3D dose distributions. The segments’ intensities are determined by solving a feasibility problem. The least-intensity feasible (LIF) solution, proposed and … Read more
The definition of a Bregman function, given by Censor and Lent in 1981 on the basis of Bregman’s seminal 1967 paper, was subsequently used in a plethora of research works as a tool for building sequential and inherently parallel feasibility and optimization algorithms. Solodov and Svaiter have recently shown that it is not CitationJournal of … Read more
The notion of relaxation is well understood for orthogonal projections onto convex sets. For general Bregman projections it was considered only for hyperplanes and the question of how to relax Bregman projections onto convex sets that are not linear (i.e., not hyperplanes or half-spaces) has remained open. A definition of underrelaxation of Bregman projections onto … Read more
We formulate a block-iterative algorithmic scheme for the solution of systems of linear inequalities and/or equations and analyze its convergence. This study provides as special cases proofs of convergence of (i) the recently proposed Component Averaging (CAV) method of Censor, Gordon and Gordon ({\it Parallel Computing}, 27:777–808, 2001), (ii) the recently proposed Block-Iterative CAV (BICAV) … Read more