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Globally Optimal Solutions to a Class of Fractional Optimization Problems Based on Proximal Gradient Algorithm

Yizun Lin, Jian-Feng Cai, Zhao-Rong Lai, Cheng Li

TL;DR

The paper addresses constrained fractional optimization where both numerator and denominator are convex and semi-algebraic, introducing a proximal gradient algorithm (PGA) that requires only a single proximal gradient step per iteration and converges to a critical point. By constructing a nonnegative surrogate F through a linear combination \\tilde{f} = f - M g and establishing equivalences between the fractional problem and its subtractive form, the authors prove global convergence to a minimizer under a mild condition f(x*) <= 0, even for unbounded constraint sets. The method is demonstrated on Sharpe ratio maximization and simple fractional models, with numerical results showing ground-truth convergence and superior performance relative to baselines and competing methods on real financial data. The work offers a practical, scalable approach to a broad class of fractional problems, supported by KL-based convergence guarantees and semi-algebraic structure that underpin convergence and rates.

Abstract

In this paper, we investigate a category of constrained fractional optimization problems that emerge in various practical applications. The objective function for this category is characterized by the ratio of a numerator and denominator, both being convex, semi-algebraic, Lipschitz continuous, and differentiable with Lipschitz continuous gradients over the constraint sets. The constrained sets associated with these problems are closed, convex, and semi-algebraic. We propose an efficient algorithm that is inspired by the proximal gradient method, and we provide a thorough convergence analysis. Our algorithm offers several benefits compared to existing methods. It requires only a single proximal gradient operation per iteration, thus avoiding the complicated inner-loop concave maximization usually required. Additionally, our method converges to a critical point without the typical need for a nonnegative numerator, and this critical point becomes a globally optimal solution with an appropriate condition. Our approach is adaptable to unbounded constraint sets as well. Therefore, our approach is viable for many more practical models. Numerical experiments show that our method not only reliably reaches ground-truth solutions in some model problems but also outperforms several existing methods in maximizing the Sharpe ratio with real-world financial data.

Globally Optimal Solutions to a Class of Fractional Optimization Problems Based on Proximal Gradient Algorithm

TL;DR

The paper addresses constrained fractional optimization where both numerator and denominator are convex and semi-algebraic, introducing a proximal gradient algorithm (PGA) that requires only a single proximal gradient step per iteration and converges to a critical point. By constructing a nonnegative surrogate F through a linear combination \\tilde{f} = f - M g and establishing equivalences between the fractional problem and its subtractive form, the authors prove global convergence to a minimizer under a mild condition f(x*) <= 0, even for unbounded constraint sets. The method is demonstrated on Sharpe ratio maximization and simple fractional models, with numerical results showing ground-truth convergence and superior performance relative to baselines and competing methods on real financial data. The work offers a practical, scalable approach to a broad class of fractional problems, supported by KL-based convergence guarantees and semi-algebraic structure that underpin convergence and rates.

Abstract

In this paper, we investigate a category of constrained fractional optimization problems that emerge in various practical applications. The objective function for this category is characterized by the ratio of a numerator and denominator, both being convex, semi-algebraic, Lipschitz continuous, and differentiable with Lipschitz continuous gradients over the constraint sets. The constrained sets associated with these problems are closed, convex, and semi-algebraic. We propose an efficient algorithm that is inspired by the proximal gradient method, and we provide a thorough convergence analysis. Our algorithm offers several benefits compared to existing methods. It requires only a single proximal gradient operation per iteration, thus avoiding the complicated inner-loop concave maximization usually required. Additionally, our method converges to a critical point without the typical need for a nonnegative numerator, and this critical point becomes a globally optimal solution with an appropriate condition. Our approach is adaptable to unbounded constraint sets as well. Therefore, our approach is viable for many more practical models. Numerical experiments show that our method not only reliably reaches ground-truth solutions in some model problems but also outperforms several existing methods in maximizing the Sharpe ratio with real-world financial data.
Paper Structure (19 sections, 26 theorems, 86 equations, 2 figures, 5 tables, 1 algorithm)

This paper contains 19 sections, 26 theorems, 86 equations, 2 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Dinkelbach's approach
  4. Pang's approach
  5. Zhang and Li's approach
  6. Proximal gradient algorithm for fractional optimization problems
  7. Development of proximal gradient algorithm
  8. Convergence analysis of PGA
  9. Numerical examples
  10. Sharpe ratio maximization model
  11. A simple example of the SR model with analytical optimal solution
  12. An example with unbounded constraint set and multiple globally optimal solutions
  13. Experimental Results
  14. Experiments for Sim1-PGA
  15. Experiments for Sim2-PGA
  16. ...and 4 more sections

Key Result

Proposition 3.2

\newlabelprop:fglbound0 Given a closed set $\Omega\subset\mathbb{R}^n$, and two continuous functions $f$ and $g$ on $\Omega$ that fulfill Assumption 2, with $g(\bm{x})>0$ for all $\bm{x}\in\Omega$, there exists a real number $M$ such that $\frac{f(\bm{x})}{g(\bm{x})}\geq M$ for all $\bm{x}\in\Omeg

Figures (2)

  • Figure 1: Experimental results of Sim1-A and Sim1-B. (a) and (b) suggest that Sim1-PGA converges to the ground-truth solutions. (c) and (d) demonstrate that Sim1-PGA monotonically decreases the objective function until it reaches its minimum. (e) and (f): By substituting $x_2=1-x_1$ into the objective function, it is transformed into a one-variable function in terms of $x_1$.
  • Figure 2: Experimental results of Sim2-A, Sim2-B, Sim2-C, and Sim2-D. (a) confirms that Sim2-PGA converges to a critical point, which is also a global minimizer of the objective function. (b) shows that Sim2-PGA monotonically reduces the objective function to its minimum. (c) presents the graph of the objective function, with its global minimizes indicated by a red line.

Theorems & Definitions (45)

  • Definition 3.1: Semi-algebraic sets and functions
  • Proposition 3.2
  • Proof 1
  • Definition 3.3: Subdifferentials and critical point
  • Lemma 3.4: Fermat's rule rockafellar2009variational
  • Lemma 3.5: rockafellar2009variational
  • Lemma 3.6: rockafellar2009variational
  • Lemma 3.7
  • Proof 2
  • Proposition 3.8
  • ...and 35 more