Adaptive Variant of the Frank-Wolfe Algorithm for Convex Optimization Problems

TL;DR

This paper introduces an adaptive variant of the Frank-Wolfe algorithm for convex optimization that automatically adjusts step parameters based on the objective function's smoothness, improving efficiency for both smooth and nonsmooth problems.

Contribution

The paper develops a new adaptive Frank-Wolfe method with theoretical guarantees and demonstrates its effectiveness through computational experiments on various problems.

Findings

The method guarantees at least a twofold reduction in function discrepancy.

It is effective for both smooth and nonsmooth convex problems.

Computational results show improved efficiency over existing algorithms.

Abstract

Some variant of the Frank-Wolfe method for convex optimization problems with adaptive selection of the step parameter corresponding to information about the smoothness of the objective function (the Lipschitz constant of the gradient). Theoretical estimates of the quality of the solution provided by the method are obtained in terms of adaptively selected parameters L_k. An important feature of the obtained result is the elaboration of a situation in which it is possible to guarantee, after the completion of the iteration, a reduction of the discrepancy in the function by at least 2 times. At the same time, using of adaptively selected parameters in theoretical estimates makes it possible to apply the method for both smooth and nonsmooth problems, provided that the exit criterion from the iteration is met. For smooth problems, this can be proved, and the theoretical estimates of the method are guaranteed to be optimal up to multiplication by a constant factor. Computational experiments were performed, and a comparison with two other algorithms was carried out, during which the efficiency of the algorithm was demonstrated for a number of both smooth and non-smooth problems.