Generalized Composed Alternating Relaxed Projection Algorithm for Two-Set Feasibility Problem
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| Format: | Preprint |
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2026
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| _version_ | 1866910145822326784 |
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| author | Li, Xinxin Wei, Yudong Zhang, Hao |
| author_facet | Li, Xinxin Wei, Yudong Zhang, Hao |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_17276 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized Composed Alternating Relaxed Projection Algorithm for Two-Set Feasibility Problem Li, Xinxin Wei, Yudong Zhang, Hao Optimization and Control Numerical Analysis 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. |
| title | Generalized Composed Alternating Relaxed Projection Algorithm for Two-Set Feasibility Problem |
| topic | Optimization and Control Numerical Analysis |
| url | https://arxiv.org/abs/2604.17276 |