A conditional gradient homotopy method with applications to Semidefinite Programming

TL;DR

This paper introduces a novel homotopy-based conditional gradient method tailored for large-scale convex optimization problems with conic constraints, especially effective in semidefinite programming relaxations of combinatorial problems.

Contribution

It presents a double-loop algorithm combining a self-concordant barrier and a conditional gradient approach, offering competitive iteration complexity with projection-free subroutines.

Findings

The method is theoretically competitive with state-of-the-art SDP solvers.

It employs cheap projection-free subroutines for efficiency.

Preliminary experiments demonstrate practical effectiveness.

Abstract

We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art SDP solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method.