On the Frank-Wolfe algorithm for non-compact constrained optimization problems

TL;DR

This paper extends the analysis of the Frank--Wolfe algorithm to a new class of non-compact constrained optimization problems characterized by specific gradient conditions, providing theoretical insights and practical examples.

Contribution

It introduces a novel class of non-compact problems for the Frank--Wolfe algorithm, characterized by gradient conditions involving the asymptotic cone, and extends classical convergence results.

Findings

Class of non-compact problems well-defined under new conditions

Extended asymptotic behavior and complexity bounds

Provided practical examples satisfying the conditions

Abstract

This paper is concerned with the Frank--Wolfe algorithm for a special class of {\it non-compact} constrained optimization problems. The notion of asymptotic cone is used to introduce this class of problems as well as to establish that the algorithm is well defined. These problems, with closed and convex constraint set, are characterized by two conditions on the gradient of the objective function. The first establishes that the gradient of the objective function is Lipschitz continuous, which is quite usual in the analysis of this algorithm. The second, which is new in this subject, establishes that the gradient belongs to the interior of the dual asymptotic cone of the constraint set. Classical results on the asymptotic behavior and iteration-complexity bounds for the sequence generated by the Frank--Wolfe algorithm are extended to this new class of problems. Examples of problems with non-compact constraints and objective functions satisfying the aforementioned conditions are also provided.