The supporting halfspace- quadratic programming strategy for the dual of the best approximation problem

TL;DR

This paper enhances Dykstra's algorithm for the best approximation problem by integrating a quadratic programming-based SHQP strategy, improving convergence rates and enabling accelerated algorithms.

Contribution

It introduces a novel SHQP strategy for the dual of BAP, extending Dykstra's algorithm with convergence guarantees and an accelerated O(1/k^2) method.

Findings

Enhanced Dykstra's algorithm with SHQP improves convergence.

Established convergence of warmstart and simultaneous variants.

Developed an accelerated O(1/k^2) dual algorithm with SHQP.

Abstract

We consider the best approximation problem (BAP) of projecting a point onto the intersection of a number of convex sets. It is known that Dykstra's algorithm is alternating minimization on the dual problem. We extend Dykstra's algorithm so that it can be enhanced by the SHQP strategy of using quadratic programming to project onto the intersection of supporting halfspaces generated by earlier projection operations. By looking at a structured alternating minimization problem, we show the convergence rate of Dykstra's algorithm when reasonable conditions are imposed to guarantee a dual minimizer. We also establish convergence of using a warmstart iterate for Dykstra's algorithm, show how all the results for the Dykstra's algorithm can be carried over to the simultaneous Dykstra's algorithm, and discuss a different way of incorporating the SHQP strategy. Lastly, we show that the dual of the best approximation problem can have an O(1/k^2) accelerated algorithm that also incorporates the SHQP strategy.