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A proximal algorithm incorporating difference of convex functions optimization for solving a class of single-ratio fractional programming

Anna Qi, Jianfeng Huang, Lihua Yang, Chao Huang

TL;DR

This work tackles single-ratio fractional minimization where both the numerator $T$ and denominator $B$ are convex and positively homogeneous. It develops PS-DCA, a proximal-subgradient-DC algorithm that combines a proximal update with a DC refinement to compute critical points, and proves subsequential convergence to critical points; under mild conditions it also guarantees global sequential convergence for a class of generalized graph Fourier mode problems. The paper derives local and global optimality conditions in the DC framework, provides a convergence theory for PS-DCA and PSA, and demonstrates practical robustness through numerical experiments on graph-based problems where the method often outperforms existing proximal-gradient-type schemes. The results advance efficient, stable solutions to fractional programming on spheres with broad applicability in signal processing and graph-based learning.

Abstract

In this paper, we consider a class of single-ratio fractional minimization problems, where both the numerator and denominator of the objective are convex functions satisfying positive homogeneity. Many nonsmooth optimization problems on the sphere that are commonly encountered in application scenarios across different scientific fields can be converted into this equivalent fractional programming. We derive local and global optimality conditions of the problem and subsequently propose a proximal-subgradient-difference of convex functions algorithm (PS-DCA) to compute its critical points. When the DCA step is removed, PS-DCA reduces to the proximal-subgradient algorithm (PSA). Under mild assumptions regarding the algorithm parameters, it is shown that any accumulation point of the sequence produced by PS-DCA or PSA is a critical point of the problem. Moreover, for a typical class of generalized graph Fourier mode problems, we establish global convergence of the entire sequence generated by PS-DCA or PSA. Numerical experiments conducted on computing the generalized graph Fourier modes demonstrate that, compared to proximal gradient-type algorithms, PS-DCA integrates difference of convex functions (d.c.) optimization, rendering it less sensitive to initial points and preventing the sequence it generates from being trapped in low-quality local minimizers.

A proximal algorithm incorporating difference of convex functions optimization for solving a class of single-ratio fractional programming

TL;DR

This work tackles single-ratio fractional minimization where both the numerator and denominator are convex and positively homogeneous. It develops PS-DCA, a proximal-subgradient-DC algorithm that combines a proximal update with a DC refinement to compute critical points, and proves subsequential convergence to critical points; under mild conditions it also guarantees global sequential convergence for a class of generalized graph Fourier mode problems. The paper derives local and global optimality conditions in the DC framework, provides a convergence theory for PS-DCA and PSA, and demonstrates practical robustness through numerical experiments on graph-based problems where the method often outperforms existing proximal-gradient-type schemes. The results advance efficient, stable solutions to fractional programming on spheres with broad applicability in signal processing and graph-based learning.

Abstract

In this paper, we consider a class of single-ratio fractional minimization problems, where both the numerator and denominator of the objective are convex functions satisfying positive homogeneity. Many nonsmooth optimization problems on the sphere that are commonly encountered in application scenarios across different scientific fields can be converted into this equivalent fractional programming. We derive local and global optimality conditions of the problem and subsequently propose a proximal-subgradient-difference of convex functions algorithm (PS-DCA) to compute its critical points. When the DCA step is removed, PS-DCA reduces to the proximal-subgradient algorithm (PSA). Under mild assumptions regarding the algorithm parameters, it is shown that any accumulation point of the sequence produced by PS-DCA or PSA is a critical point of the problem. Moreover, for a typical class of generalized graph Fourier mode problems, we establish global convergence of the entire sequence generated by PS-DCA or PSA. Numerical experiments conducted on computing the generalized graph Fourier modes demonstrate that, compared to proximal gradient-type algorithms, PS-DCA integrates difference of convex functions (d.c.) optimization, rendering it less sensitive to initial points and preventing the sequence it generates from being trapped in low-quality local minimizers.
Paper Structure (12 sections, 12 theorems, 112 equations, 5 tables, 3 algorithms)

This paper contains 12 sections, 12 theorems, 112 equations, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Local and global optimality conditions for Problem $P_1$
  4. The proximal-subgradient-d.c. algorithm (PS-DCA) for solving Problem $P_1(P_2)$
  5. Convergence analysis of PS-DCA
  6. Application to the generalized graph Fourier mode problem
  7. Global sequential convergence of PS-DCA or PSA
  8. Choosing the subgradient $w^k$ in PS-DCA for problem \ref{['GFM-fracprogram']}
  9. Numerical Experiments
  10. Conclusion
  11. Proof of the equivalence between problem \ref{['eq:P1']} and fractional programming \ref{['eq:P2']}
  12. Proof of Proposition \ref{['loc trans prop']}

Key Result

Theorem 2.1

Suppose that the sequence $\{x^k\}$ and $\{y^k\}$ are generated by the simplified DCA. Then we have where $0\leq\rho_i<\rho(f_i)$$(resp.~0\leq\rho^*_i<\rho(f^*_i)),(i=1,2)$, and $\rho_i=0$$(resp. ~\rho^*_i=0)$ if $\rho(f_i)=0$$(resp. ~\rho(f_i^*)=0)$. Specifically, if the supremum $\rho(f_i)$$(resp. ~\rho(f_i^*))$ is attainable, then $\rho_i$$(resp. ~\rho^*_i)$ may take this value. $dx^k:=x^{k+1}

Theorems & Definitions (16)

  • Definition 2.1
  • Theorem 2.1
  • Definition 3.1
  • Proposition 3.1
  • Definition 3.2
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Lemma 5.1: Monotonicity of PS-DCA
  • Lemma 5.2: Compactness of PS-DCA
  • ...and 6 more