Generalized Composed Alternating Relaxed Projection Algorithm for Two-Set Feasibility Problem

TL;DR

This paper introduces a generalized algorithm for the two-set feasibility problem in Hilbert spaces, blending existing methods with new relaxation and averaging techniques, and demonstrates its convergence and effectiveness through spectral analysis and experiments.

Contribution

It proposes a novel generalized composed alternating relaxed projection algorithm (gCARPA) that unifies and extends classical projection methods, including a non-stationary variant with proven convergence.

Findings

The spectral characterization provides explicit eigenvalue bounds for the subspace model.

Non-stationary tuning of relaxation parameters can enhance convergence.

Numerical experiments show improved or comparable performance to baseline methods.

Abstract

We study the two-set feasibility problem of finding a point in the intersection $X\cap Y$ of closed convex sets in a Hilbert space. We propose a generalized composed alternating relaxed projection algorithm (gCARPA) that blends Douglas-Rachford-type and projection-reflection-type dynamics via an outer averaging step $\mu$ and an internal relaxation $(\gamma,\theta,\eta)$. The algorithm contains several classical projection methods as special cases. We also introduce its non-stationary variant, in which $(\gamma_k,\theta_k,\eta_k)$ vary over iterations, and establish its convergence. For the subspace feasibility model, we derive an explicit spectral characterization via principal-angle block decompositions, yielding computable subdominant-eigenvalue factors and a minimax parameter-selection recipe in a symmetric regime that targets critical damping on principal-angle planes. Numerical experiments illustrate that the generalized relaxation and its non-stationary tuning can improve or match baseline methods in problem-dependent regimes.