Exterior-Point Optimization for Sparse and Low-Rank Optimization

Shuvomoy Das Gupta (), Bartolomeo Stellato () and Bart P. G. Parys ()
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Shuvomoy Das Gupta: Massachusetts Institute of Technology
Bartolomeo Stellato: Princeton University
Bart P. G. Parys: Massachusetts Institute of Technology

Journal of Optimization Theory and Applications, 2024, vol. 202, issue 2, No 11, 795-833

Abstract: Abstract Many problems of substantial current interest in machine learning, statistics, and data science can be formulated as sparse and low-rank optimization problems. In this paper, we present the nonconvex exterior-point optimization solver (NExOS)—a first-order algorithm tailored to sparse and low-rank optimization problems. We consider the problem of minimizing a convex function over a nonconvex constraint set, where the set can be decomposed as the intersection of a compact convex set and a nonconvex set involving sparse or low-rank constraints. Unlike the convex relaxation approaches, NExOS finds a locally optimal point of the original problem by solving a sequence of penalized problems with strictly decreasing penalty parameters by exploiting the nonconvex geometry. NExOS solves each penalized problem by applying a first-order algorithm, which converges linearly to a local minimum of the corresponding penalized formulation under regularity conditions. Furthermore, the local minima of the penalized problems converge to a local minimum of the original problem as the penalty parameter goes to zero. We then implement and test NExOS on many instances from a wide variety of sparse and low-rank optimization problems, empirically demonstrating that our algorithm outperforms specialized methods.

Keywords: Nonconvex optimization; Sparse optimization; Low-rank optimization; First-order algorithms; 65K05; 90C30 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02448-9

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