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

TL;DR

This paper introduces a novel proximal algorithm, PS-DCA, for solving single-ratio fractional programming problems with convex, positively homogeneous numerator and denominator, demonstrating improved convergence and robustness over existing methods.

Contribution

The paper develops the PS-DCA algorithm that incorporates difference of convex functions optimization, providing convergence guarantees and better performance in fractional programming.

Findings

PS-DCA converges to critical points under mild conditions.

Compared to proximal gradient methods, PS-DCA is less sensitive to initial points.

Numerical experiments show PS-DCA effectively computes generalized graph Fourier modes.

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.