Generalized Composed Alternating Relaxed Projection Algorithm for Two-Set Feasibility Problem
This work provides a unified algorithmic framework for two-set feasibility problems, offering flexibility through relaxation parameters, but the improvements are problem-dependent and incremental over existing methods.
The paper proposes a generalized composed alternating relaxed projection algorithm (gCARPA) for finding a point in the intersection of two closed convex sets in Hilbert space, blending Douglas-Rachford-type and projection-reflection-type dynamics. The algorithm includes classical methods as special cases and introduces a non-stationary variant with convergence guarantees; numerical experiments show that the generalized relaxation can improve or match baseline methods in problem-dependent regimes.
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 $μ$ and an internal relaxation $(γ,θ,η)$. The algorithm contains several classical projection methods as special cases. We also introduce its non-stationary variant, in which $(γ_k,θ_k,η_k)$ 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.