Year: 2025
Type: preprint
Abstract: Basis pursuit is the problem of finding a vector with smallest $\ell_1$-norm among the solutions of a given linear system of equations. It is a well-known convex relaxation of the sparse affine feasibility problem, where sparse solutions to underdetermined systems are sought. Since basis pursuit admits a linear programming reformulation, standard LP solvers are directly applicable. We instead address the basis pursuit directly in its $\ell_1$-minimization form, without LP reformulation, via a scheme that uses alternating projections in its subproblems. These subproblems are designed to be inconsistent in the sense that they relate to two non-intersecting sets. Recently in [R. Behling, Y. Bello-Cruz and L.-R. Santos, SIAM J. Optim., 31 (2021), pp. 2863-2892], inconsistency coming from infeasibility has been shown to accelerate convergence of alternating projections. We deliberately enforce this inconsistency by constructing subproblems whose feasible sets are disjoint by design. We prove that the resulting $\ell_1$-radii converge linearly to the optimal value, and that when the solution is unique, all generated sequences converge linearly to it at a rate governed by a natural error bound between the feasible set and the optimal $\ell_1$-ball. The proposed method is numerically competitive against state-of-the-art open-source solvers on synthetic and real-world instances.
Source: arXiv (Cornell University)
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Funders: National Science Foundation, Fundação de Amparo à Pesquisa do Estado de São Paulo, Conselho Nacional de Desenvolvimento Científico e Tecnológico, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina
OpenAlex ID: W4416050365
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