Constraint reduction reformulations for projection algorithms with applications to wavelet construction

TL;DR

This paper introduces a constraint reduction reformulation technique that simplifies feasibility problems, enhances projection algorithms, and demonstrates improved performance in wavelet construction applications.

Contribution

The paper presents a novel constraint reduction reformulation that reduces problem dimension and extends convergence analysis for projection algorithms, applied to wavelet feasibility problems.

Findings

Constraint reduction improves algorithm efficiency.

Global convergence in convex settings is established.

Enhanced performance in wavelet construction tasks.

Abstract

We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra's classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas--Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyse convergence of the constraint-reduced Douglas--Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas--Rachford and alternating projections to solve the wavelet feasibility problems, and then compare their performance with their usual product variants.