Inertial Proximal Block Coordinate Method for a Class of Nonsmooth Sum-of-Ratios Optimization Problems

TL;DR

This paper introduces an inertial proximal block coordinate method for nonsmooth sum-of-ratios fractional optimization problems, demonstrating its convergence and linear rate for certain structured problems, supported by theoretical and numerical evidence.

Contribution

The paper develops a novel inertial proximal block coordinate algorithm with proven convergence for a broad class of nonsmooth sum-of-ratios problems, including explicit KL exponent analysis.

Findings

Method guarantees global convergence under KL property.

Linear convergence rate for sparse generalized eigenvalue problems.

Validated effectiveness through numerical experiments.

Abstract

In this paper, we consider a class of nonsmooth sum-of-ratios fractional optimization problems with block structure. This model class is ubiquitous and encompasses several important nonsmooth optimization problems in the literature. We first propose an inertial proximal block coordinate method for solving this class of problems by exploiting the underlying structure. The global convergence of our method is guaranteed under the Kurdyka--Lojasiewicz (KL) property and some mild assumptions. We then identify the explicit exponents of the KL property for three important structured fractional optimization problems. In particular, for the sparse generalized eigenvalue problem with either cardinality regularization or sparsity constraint, we show that the KL exponents are 1/2, and so, the proposed method exhibits linear convergence rate. Finally, we illustrate our theoretical results with both analytic and simulated numerical examples.