On fast convergence rates for generalized conditional gradient methods with backtracking stepsize

TL;DR

This paper introduces a generalized conditional gradient method with backtracking stepsize for convex optimization, achieving improved convergence rates without requiring strong convexity, and demonstrates its efficiency on PDE-constrained problems.

Contribution

The paper presents a new conditional gradient algorithm with backtracking stepsize that attains faster convergence rates under mild conditions, without needing strong convexity.

Findings

Sequences converge to minimizers under mild assumptions

Achieves sublinear convergence rates for objective values

Numerical tests confirm practical efficiency on PDE problems

Abstract

A generalized conditional gradient method for minimizing the sum of two convex functions, one of them differentiable, is presented. This iterative method relies on two main ingredients: First, the minimization of a partially linearized objective functional to compute a descent direction and, second, a stepsize choice based on an Armijo-like condition to ensure sufficient descent in every iteration. We provide several convergence results. Under mild assumptions, the method generates sequences of iterates which converge, on subsequences, towards minimizers. Moreover a sublinear rate of convergence for the objective functional values is derived. Second, we show that the method enjoys improved rates of convergence if the partially linearized problem fulfills certain growth estimates. Most notably these results do not require strong convexity of the objective functional. Numerical tests on a variety of challenging PDE-constrained optimization problems confirm the practical efficiency of the proposed algorithm.