Set intersection problems: Integrating projection and quadratic programming algorithms

TL;DR

This paper explores integrating projection algorithms with active set quadratic programming to improve convergence in set intersection problems, including box constraints and algebraic reconstruction, highlighting the warmstart property benefits.

Contribution

It introduces a novel integration of projection methods with active set QP algorithms, enhancing convergence speed and extending existing algorithms for convex set intersections.

Findings

Accelerates convergence for set intersection algorithms.

Extends Algebraic Reconstruction Technique with finite convergence.

Highlights the warmstart property of active set QP 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.