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

Deren Han, Min Tao, Zihao Xia

TL;DR

The authors address nonconvex and nonsmooth fractional programs of the form $\min_{\mathbf{x}\in \mathcal{S}} F(\mathbf{x})$ with $F(\mathbf{x}) = \dfrac{g(\mathbf{x}) + h(\mathbf{x})}{f(K\mathbf{x})}$, where the numerator is a sum of a convex nonsmooth and a differentiable nonconvex term and the denominator is a convex nonsmooth term composed with a linear operator. They develop a single-loop fully-splitting proximal subgradient algorithm (FPSA) with a relaxation step, and a nonmonotone variant FPSA-nl, and prove subsequential and global convergence to an exact lifted stationary point under Kurdyka-\Lojasiewicz (KL) assumptions. The work extends prior results by decoupling the linear operator from the nonsmooth part and by ensuring exact lifted stationarity in the presence of a linear operator in the denominator, even for nonconvex numerators. Numerical experiments on $L_{1}/S_{\kappa}$ sparse recovery, limited-angle CT reconstruction, and single-period portfolio optimization show that FPSA-nl outperforms state-of-the-art methods in speed and accuracy, highlighting its practical impact for high-dimensional, structured fractional programs.

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.

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

TL;DR

The authors address nonconvex and nonsmooth fractional programs of the form with , where the numerator is a sum of a convex nonsmooth and a differentiable nonconvex term and the denominator is a convex nonsmooth term composed with a linear operator. They develop a single-loop fully-splitting proximal subgradient algorithm (FPSA) with a relaxation step, and a nonmonotone variant FPSA-nl, and prove subsequential and global convergence to an exact lifted stationary point under Kurdyka-\Lojasiewicz (KL) assumptions. The work extends prior results by decoupling the linear operator from the nonsmooth part and by ensuring exact lifted stationarity in the presence of a linear operator in the denominator, even for nonconvex numerators. Numerical experiments on sparse recovery, limited-angle CT reconstruction, and single-period portfolio optimization show that FPSA-nl outperforms state-of-the-art methods in speed and accuracy, highlighting its practical impact for high-dimensional, structured fractional programs.

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: sparse signal recovery, limited-angle CT reconstruction, and optimal portfolio selection.

Paper Structure

This paper contains 13 sections, 10 theorems, 90 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed algorithm
  4. Fully-splitting proximal subgradient
  5. Subsequential convergence
  6. Global convergence
  7. fstar satisfies the calm condition
  8. f is ${\cal C}^1$ over an open set containing KS
  9. Numerical Results
  10. $L_{1}/S_{\kappa}$ sparse signal recovery
  11. Limited-Angle CT Reconstruction
  12. Optimal Portfolio Selection
  13. Conclusions

Key Result

lemma 1

boct2023full Let $O \subseteq \mathbb{R}^n$ be an open set, and $f_1 : O \rightarrow \overline{\mathbb \mathbb{R}}$ and $f_2 : O \rightarrow \mathbb \mathbb{R}$ be two functions which are finite at ${\mathbf{x}} \in O$ with $f_2({\mathbf{x}})>0$. Suppose that $f_1$ is continuous at ${\mathbf{x}}$ re (ii) If, in addition, $f_2$ is convex and $\alpha_1\ge0$, then

Figures (5)

  • Figure 1: The averaged results of CPU time from three algorithms with $\kappa\in\{4, 8, 12\}$ and $F\in\{1, 5, 10, 15, 20\}$.
  • Figure 2: The averaged results of StatRes from three algorithms with $\kappa\in\{4, 8, 12\}$ and $F\in\{1, 5, 10, 15, 20\}$.
  • Figure 3: The averaged results of Err from three algorithms with $\kappa\in\{4, 8, 12\}$ and $F\in\{1, 5, 10, 15, 20\}$.
  • Figure 4: The original images of FB, as well as the reconstructed results from FPSA-nl, PGSA_BE, e-PSG (from left to right) under the projection angle of 150$^{\circ}$ and noise level at 0.1%.
  • Figure 5: The performance profile of CPU time and StatRes.

Theorems & Definitions (27)

  • definition 1
  • definition 2
  • lemma 1
  • lemma 2
  • proof
  • remark 1
  • definition 3
  • remark 2
  • theorem 3
  • proof
  • ...and 17 more