A Single-loop Proximal Subgradient Algorithm for A Class Structured Fractional Programs
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| Format: | Preprint |
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2025
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| author | Han, Deren Tao, Min Xia, Zihao |
| author_facet | Han, Deren Tao, Min Xia, Zihao |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Single-loop Proximal Subgradient Algorithm for A Class Structured Fractional Programs Han, Deren Tao, Min Xia, Zihao Optimization and Control 90C26, 90C32, 49M27, 65K05 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. |
| title | A Single-loop Proximal Subgradient Algorithm for A Class Structured Fractional Programs |
| topic | Optimization and Control 90C26, 90C32, 49M27, 65K05 |
| url | https://arxiv.org/abs/2503.12176 |