Set intersection problems: Integrating projection and quadratic programming algorithms

Abstract

Abstract. The Set Intersection Problem (SIP) is the problem of finding a point in the intersection of convex sets. This problem is typically solved by the method of alternating projections. To accelerate the convergence, the idea of using Quadratic Programming (QP) to project a point onto the intersection of halfspaces generated by the projection process was discussed in earlier papers. This paper looks at how one can integrate projection algorithms together with an active set QP algorithm. As a byproduct of our analysis, we show how to accelerate an SIP algorithm involving box constraints, and how to extend a version of the Algebraic Reconstruction Technique (ART) while preserving finite convergence. Lastly, the warmstart property of active set QP algorithms is a valuable property for the problem of projecting onto the intersection of convex sets.