Proximal-gradient algorithms for fractional programming

TL;DR

This paper introduces two proximal gradient algorithms for fractional programming in Hilbert spaces, addressing both concave and convex denominators, with convergence guarantees to optimal solutions or critical points.

Contribution

It proposes novel proximal gradient methods tailored for fractional programming with specific convergence properties in Hilbert spaces.

Findings

Algorithms converge to global solutions for concave denominators.

Algorithms approach critical points for convex denominators under Kurdyka-iewicz property.

The methods extend fractional programming solution techniques in infinite-dimensional spaces.

Abstract

In this paper we propose two proximal gradient algorithms for fractional programming problems in real Hilbert spaces, where the numerator is a proper, convex and lower semicontinuous function and the denominator is a smooth function, either concave or convex. In the iterative schemes, we perform a proximal step with respect to the nonsmooth numerator and a gradient step with respect to the smooth denominator. The algorithm in case of a concave denominator has the particularity that it generates sequences which approach both the (global) optimal solutions set and the optimal objective value of the underlying fractional programming problem. In case of a convex denominator the numerical scheme approaches the set of critical points of the objective function, provided the latter satisfies the Kurdyka-\L{}ojasiewicz property.