A Single-loop Proximal Subgradient Algorithm for A Class Structured Fractional Programs

In this paper, we investigate a class of nonconvex and nonsmooth fractional programming problems, where the numerator composed of two parts: a convex, nonsmooth function and a differentiable, nonconvex function, and the denominator consists of a convex, nonsmooth function composed of a linear operat...

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Published in:	Journal of scientific computing Vol. 104; no. 3; p. 75
Main Authors:	Han, Deren, Tao, Min, Xia, Zihao
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.09.2025 Springer Nature B.V
Subjects:	Algorithms Computational Mathematics and Numerical Analysis Decoupling Linear operators Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical programming Mathematics Mathematics and Statistics Optimization Recovery Signal reconstruction Theoretical Decoupling Fractional programming Single-loop Convergence analysis
ISSN:	0885-7474, 1573-7691
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Abstract	In this paper, we investigate a class of nonconvex and nonsmooth fractional programming problems, where the numerator composed of two parts: a convex, nonsmooth function and a differentiable, nonconvex function, and the denominator consists of a convex, nonsmooth function composed of a linear operator. These structured fractional programming problems have broad applications, including CT reconstruction, sparse signal recovery, the single-period optimal portfolio selection problem and standard Sharpe ratio minimization problem. We develop a single-loop proximal subgradient algorithm that alleviates computational complexity by decoupling the evaluation of the linear operator from the nonsmooth component. We prove the global convergence of the proposed single-loop algorithm to an exact lifted stationary point under the Kurdyka-Łojasiewicz assumption. Additionally, we present a practical variant incorporating a nonmonotone line search to improve computational efficiency. Finally, through extensive numerical simulations, we showcase the superiority of the proposed approach over the existing state-of-the-art methods for three applications: L 1 / S κ sparse signal recovery, limited-angle CT reconstruction, and optimal portfolio selection.
AbstractList	In this paper, we investigate a class of nonconvex and nonsmooth fractional programming problems, where the numerator composed of two parts: a convex, nonsmooth function and a differentiable, nonconvex function, and the denominator consists of a convex, nonsmooth function composed of a linear operator. These structured fractional programming problems have broad applications, including CT reconstruction, sparse signal recovery, the single-period optimal portfolio selection problem and standard Sharpe ratio minimization problem. We develop a single-loop proximal subgradient algorithm that alleviates computational complexity by decoupling the evaluation of the linear operator from the nonsmooth component. We prove the global convergence of the proposed single-loop algorithm to an exact lifted stationary point under the Kurdyka-Łojasiewicz assumption. Additionally, we present a practical variant incorporating a nonmonotone line search to improve computational efficiency. Finally, through extensive numerical simulations, we showcase the superiority of the proposed approach over the existing state-of-the-art methods for three applications: L 1 / S κ sparse signal recovery, limited-angle CT reconstruction, and optimal portfolio selection. In this paper, we investigate a class of nonconvex and nonsmooth fractional programming problems, where the numerator composed of two parts: a convex, nonsmooth function and a differentiable, nonconvex function, and the denominator consists of a convex, nonsmooth function composed of a linear operator. These structured fractional programming problems have broad applications, including CT reconstruction, sparse signal recovery, the single-period optimal portfolio selection problem and standard Sharpe ratio minimization problem. We develop a single-loop proximal subgradient algorithm that alleviates computational complexity by decoupling the evaluation of the linear operator from the nonsmooth component. We prove the global convergence of the proposed single-loop algorithm to an exact lifted stationary point under the Kurdyka-Łojasiewicz assumption. Additionally, we present a practical variant incorporating a nonmonotone line search to improve computational efficiency. Finally, through extensive numerical simulations, we showcase the superiority of the proposed approach over the existing state-of-the-art methods for three applications: L1/Sκ sparse signal recovery, limited-angle CT reconstruction, and optimal portfolio selection.
ArticleNumber	75
Author	Tao, Min Xia, Zihao Han, Deren
Author_xml	– sequence: 1 givenname: Deren surname: Han fullname: Han, Deren organization: School of Mathematical Sciences, Beihang University – sequence: 2 givenname: Min orcidid: 0000-0003-4750-2959 surname: Tao fullname: Tao, Min email: taom@nju.edu.cn organization: School of Mathematics, National Key Laboratory for Novel Software Technology, Nanjing University – sequence: 3 givenname: Zihao surname: Xia fullname: Xia, Zihao organization: School of Mathematics, National Key Laboratory for Novel Software Technology, Nanjing University
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Cites_doi	10.1007/978-3-642-02431-3 10.1137/050644641 10.1080/02331934.2017.1294592 10.1088/0031-9155/57/13/N237 10.1016/j.acha.2021.08.004 10.1007/s11075-018-0570-7 10.1287/moor.1100.0449 10.1287/opre.28.4.927 10.1137/20M1325381 10.1007/s10107-011-0484-9 10.1007/s10107-018-1292-2 10.1007/BF02591871 10.1137/1.9781611974997 10.1287/moor.2021.1214 10.1016/j.cam.2011.09.039 10.1515/9781400873173 10.1287/mnsc.13.7.492 10.1007/978-1-4615-2025-2_10 10.1137/20M136801X 10.1080/14697680903081881 10.1137/20M1355380 10.1137/22M1472000 10.1287/moor.2024.0457 10.1137/20M1341490 10.1137/1.9781611976748
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SubjectTerms	Algorithms Computational Mathematics and Numerical Analysis Decoupling Linear operators Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical programming Mathematics Mathematics and Statistics Optimization Recovery Signal reconstruction Theoretical
Title	A Single-loop Proximal Subgradient Algorithm for A Class Structured Fractional Programs
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