Line Search For Generalized Alternating Projections

TL;DR

This paper introduces a line search method for the generalized alternating projections (GAP) algorithm, improving convergence and efficiency for convex optimization problems through theoretical analysis and numerical experiments.

Contribution

It adapts a line search technique to GAP, proves its convergence, and demonstrates significant performance improvements over traditional methods.

Findings

Line search can be performed with minimal additional cost

The projected line search converges and has convex step length conditions

Numerical results show orders of magnitude speedup

Abstract

This paper is about line search for the generalized alternating projections (GAP) method. This method is a generalization of the von Neumann alternating projections method, where instead of performing alternating projections, relaxed projections are alternated. The method can be interpreted as an averaged iteration of a nonexpansive mapping. Therefore, a recently proposed line search method for such algorithms is applicable to GAP. We evaluate this line search and show situations when the line search can be performed with little additional cost. We also present a variation of the basic line search for GAP - the projected line search. We prove its convergence and show that the line search condition is convex in the step length parameter. We show that almost all convex optimization problems can be solved using this approach and numerical results show superior performance with both the standard and the projected line search, sometimes by several orders of magnitude, compared to the nominal method.