Finding the projection of a point onto the intersection of convex sets via projections onto half-spaces

• Published in: Journal of approximation theory Vol. 124; no. 2; pp. 194 - 218
• Main Authors: Bregman, Lev M., Censor, Yair, Reich, Simeon, Zepkowitz-Malachi, Yael
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.10.2003
• Subjects: Convex set; Dykstra's algorithm; Half-space; Projection; Projection; Convex set; Dykstra's algorithm; Half-space
• ISSN: 0021-9045; 1096-0430
• DOI: 10.1016/j.jat.2003.08.004

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 half-space or onto the intersection of two half-spaces. Convergence of the algorithm is established and special choices of the half-spaces are proposed. The option to project onto half-spaces instead of general convex sets makes the algorithm more practical. The fact that the half-spaces are quite general enables us to apply the algorithm in a variety of cases and to generalize a number of known projection algorithms. The problem of projecting a point onto the intersection of closed convex sets receives considerable attention in many areas of mathematics and physics as well as in other fields of science and engineering such as image reconstruction from projections. In this work we propose a new class of algorithms which allow projection onto certain super half-spaces, i.e., half-spaces which contain the convex sets. Each one of the algorithms that we present gives the user freedom to choose the specific super half-space from a family of such half-spaces. Since projecting a point onto a half-space is an easy task to perform, the new algorithms may be more useful in practical situations in which the construction of the super half-spaces themselves is not too difficult.