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A full splitting algorithm for fractional programs with structured numerators and denominators

Radu Ioan Boţ, Guoyin Li, Min Tao

TL;DR

This work tackles a broad class of nonconvex, nonsmooth fractional programs with structured numerators and denominators by proposing a single-loop, fully splitting proximal subgradient algorithm (Adaptive FSPS). The method leverages conjugate unfolding to decouple linear compositions in the numerator, applies forward steps for the linear operators, and uses an adaptive scheme to maintain positivity of a merit function while updating proximal and extrapolated components. Under KL assumptions and reasonable regularity, the authors establish subsequential convergence to approximate lifted stationary points for nonsmooth cases and global convergence to exact lifted stationary points when the smoothness conditions hold, without requiring full-row rank on the linear operators. They also introduce a nonmonotone line-search variant (Adaptive FSPS-nls) and demonstrate superior performance on limited-angle CT reconstruction and robust Sharpe ratio minimization relative to state-of-the-art double-loop methods, highlighting practical impact for structured fractional programs. The work provides rigorous convergence theory, practical algorithmic refinements, and motivating counterexamples to clarify the limits of exact stationary convergence in the nonsmooth setting.

Abstract

In this paper, we consider a class of nonconvex and nonsmooth fractional programming problems, that involve the sum of a convex, possibly nonsmooth function composed with a linear operator and a differentiable, possibly nonconvex function in the numerator and a convex, possibly nonsmooth function composed with a linear operator in the denominator. These problems have applications in various fields. We propose an adaptive full-splitting proximal subgradient algorithm that addresses the challenge of decoupling the composition of the nonsmooth component with the linear operator in the numerator. We specifically evaluate the nonsmooth function in the numerator using its proximal operator of its conjugate function. Furthermore, the smooth component in the numerator is evaluated through its gradient, and the nonsmooth in the denominator is managed using its subgradient. We demonstrate subsequential convergence toward an approximate lifted stationary point and ensure global convergence under the Kurdyka-Łojasiewicz property, all achieved without full-row rank assumptions on the linear operators. We provide further discussions on {\it the tightness of the convergence results of the proposed algorithm and its related variants, and the reasoning behind aiming for an approximate lifted stationary point}. We construct a series of counter-examples to show that the proposed algorithm and its variant might diverge when seeking exact solutions. A practical version incorporating a nonmonotone line search is also developed to enhance its performance significantly. Our theoretical findings are validated through simulations involving limited-angle CT reconstruction and the robust sharp-ratio-type minimization problem.

A full splitting algorithm for fractional programs with structured numerators and denominators

TL;DR

This work tackles a broad class of nonconvex, nonsmooth fractional programs with structured numerators and denominators by proposing a single-loop, fully splitting proximal subgradient algorithm (Adaptive FSPS). The method leverages conjugate unfolding to decouple linear compositions in the numerator, applies forward steps for the linear operators, and uses an adaptive scheme to maintain positivity of a merit function while updating proximal and extrapolated components. Under KL assumptions and reasonable regularity, the authors establish subsequential convergence to approximate lifted stationary points for nonsmooth cases and global convergence to exact lifted stationary points when the smoothness conditions hold, without requiring full-row rank on the linear operators. They also introduce a nonmonotone line-search variant (Adaptive FSPS-nls) and demonstrate superior performance on limited-angle CT reconstruction and robust Sharpe ratio minimization relative to state-of-the-art double-loop methods, highlighting practical impact for structured fractional programs. The work provides rigorous convergence theory, practical algorithmic refinements, and motivating counterexamples to clarify the limits of exact stationary convergence in the nonsmooth setting.

Abstract

In this paper, we consider a class of nonconvex and nonsmooth fractional programming problems, that involve the sum of a convex, possibly nonsmooth function composed with a linear operator and a differentiable, possibly nonconvex function in the numerator and a convex, possibly nonsmooth function composed with a linear operator in the denominator. These problems have applications in various fields. We propose an adaptive full-splitting proximal subgradient algorithm that addresses the challenge of decoupling the composition of the nonsmooth component with the linear operator in the numerator. We specifically evaluate the nonsmooth function in the numerator using its proximal operator of its conjugate function. Furthermore, the smooth component in the numerator is evaluated through its gradient, and the nonsmooth in the denominator is managed using its subgradient. We demonstrate subsequential convergence toward an approximate lifted stationary point and ensure global convergence under the Kurdyka-Łojasiewicz property, all achieved without full-row rank assumptions on the linear operators. We provide further discussions on {\it the tightness of the convergence results of the proposed algorithm and its related variants, and the reasoning behind aiming for an approximate lifted stationary point}. We construct a series of counter-examples to show that the proposed algorithm and its variant might diverge when seeking exact solutions. A practical version incorporating a nonmonotone line search is also developed to enhance its performance significantly. Our theoretical findings are validated through simulations involving limited-angle CT reconstruction and the robust sharp-ratio-type minimization problem.
Paper Structure (18 sections, 14 theorems, 82 equations, 1 figure, 1 table)

This paper contains 18 sections, 14 theorems, 82 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and calculus rules
  3. Basic assumptions and stationary points of fractional programs
  4. Full splitting adaptive algorithm
  5. Conceptual algorithmic framework
  6. Adaptive FSPS algorithm with extrapolation
  7. Convergence analysis of the Adaptive FSPS algorithm
  8. Convergence of the full sequence
  9. fstar satisfies the calm condition over its effective domain and g is essentially strictly convex
  10. f is differentiable with Lipschitz continuous gradient over an open set containing KS and g is essentially strictly convex
  11. Global convergence
  12. Further discussions
  13. Numerical results
  14. Implementation details
  15. Limited-angle CT reconstruction
  16. ...and 3 more sections

Key Result

Lemma 2.3

\newlabelratioC20 Let $O \subseteq \mathop{\mathrm{\mathbb{R}}}\nolimits^n$ be an open set, and $f_1 : O \rightarrow \overline{\mathbb \mathop{\mathrm{\mathbb{R}}}\nolimits}$ and $f_2 : O \rightarrow \mathbb \mathop{\mathrm{\mathbb{R}}}\nolimits$ be two functions which are finite at ${\mathbf{x}} (ii) If, in addition, $f_2$ is convex and $\alpha_1\ge0$, then

Figures (1)

  • Figure 1: The evolution of the relative error (left) and the objective function value (right)

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Proof 1
  • Lemma 2.4
  • Proof 2
  • Remark 3.2
  • Definition 3.3
  • Example 3.4
  • Definition 3.5
  • ...and 33 more