A Composed Alternating Relaxed Projection Algorithm for Feasibility Problem

TL;DR

The paper tackles convex feasibility problems of finding a common point in two closed convex sets $X$ and $Y$ by introducing CARPA, a composed alternating relaxed projection algorithm that blends a Douglas--Rachford–type step with a projection–reflection component, plus a non-stationary variant nsCARPA. It provides convergence guarantees by expressing the method as a fixed-point iteration with operators $\mathcal{F}_{\gamma}$ and $\mathcal{F}_{\gamma,\mu}$, and proves that the fixed-point set equals $X\cap Y$. In the two-subspace setting, the authors derive linear convergence rates and identify nearly optimal parameter choices, supported by extensive numerical experiments on subspace feasibility, a ball-vs-line problem, and sparse linear inverse problems. The results show CARPA and especially nsCARPA offer competitive or superior practical performance across various geometries, confirming the method’s potential to accelerate projection-based feasibility solvers in practice.

Abstract

Feasibility problem aims to find a common point of two or more closed (convex) sets whose intersection is nonempty. In the literature, projection based algorithms are widely adopted to solve the problem, such as the method of alternating projection (MAP), and Douglas--Rachford splitting method (DR). The performance of the methods are governed by the geometric properties of the underlying sets. For example, the fixed-point sequence of the Douglas--Rachford splitting method exhibits a spiraling behavior when solving the feasibility problem of two subspaces, leading to a slow convergence speed and slower than MAP. However, when the problem at hand is non-polyhedral, DR can demonstrate significant faster performance. Motivated by the behaviors of the DR method, in this paper we propose a new algorithm for solving convex feasibility problems. The method is designed based on DR method by further incorporating a composition of projection and reflection. A non-stationary version of the method is also designed, aiming to achieve faster practical performance. Theoretical guarantees of the proposed schemes are provided and supported by numerical experiments.

Figures (5)

• Figure 1: Spiraling behavior of the fixed-point sequence of Douglas--Rachford splitting method.
• Figure 2: Optimal convergence rates of several representative methods.
• Figure 3: Experimental results for different parameter sets.
• Figure 4: Comparison on toy example: (a) Convergence of ${||} {z}^{(k)}-{z}^\star {||}$; (b) Change of the support of $\mathscr{P}_{Y}({z}^{(k)})$.
• Figure 5: Experimental results for different problems.

Theorems & Definitions (17)

• Definition 2.1: Nonexpansive operator
• Definition 2.2: Projection
• Definition 2.3: Relaxed projection and reflection
• Definition 2.4: Principal angles
• Definition 2.5: Friedrichs angle
• Proposition 2.6: Principal angles and Friedrichs angle
• Theorem 2.8: bauschke2016optimal
• Proposition 3.1
• Proposition 3.2: Set of fixed-points
• Proposition 3.3: Convergence of CARPA