Accelerated Inexact Composite Gradient Methods for Nonconvex Spectral Optimization Problems

TL;DR

This paper introduces two accelerated inexact composite gradient methods tailored for nonconvex spectral optimization problems, leveraging spectral structure for efficiency and demonstrating practical effectiveness through numerical experiments.

Contribution

The paper develops novel accelerated inexact composite gradient algorithms that exploit spectral structure in nonconvex optimization, improving efficiency over existing methods.

Findings

Methods effectively solve nonconvex spectral problems.

Algorithms outperform traditional approaches in experiments.

Numerical results confirm practical applicability.

Abstract

This paper presents two inexact composite gradient methods, one inner accelerated and another doubly accelerated, for solving a class of nonconvex spectral composite optimization problems. More specifically, the objective function for these problems is of the form $f_1 + f_2 + h$ where $f_1$ and $f_2$ are differentiable nonconvex matrix functions with Lipschitz continuous gradients, $h$ is a proper closed convex matrix function, and both $f_2$ and $h$ can be expressed as functions that operate on the singular values of their inputs. The methods essentially use an accelerated composite gradient method to solve a sequence of proximal subproblems involving the linear approximation of $f_1$ and the singular value functions underlying $f_2$ and $h$. Unlike other composite gradient-based methods, the proposed methods take advantage of both the composite and spectral structure underlying the objective function in order to efficiently generate their solutions. Numerical experiments are presented to demonstrate the practicality of these methods on a set of real-world and randomly generated spectral optimization problems.