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

arXiv: https://arxiv.org/abs/2101.08830 · HTML: https://mlsanity.com/paper/2101.08830/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jan 2021.

## Built on by (most cited first, 2 of 2)

- [Self-adaptive algorithms for quasiconvex programming and applications to machine learning](https://mlsanity.com/paper/2212.06379/) (2022)
- [Adaptive multi-gradient methods for quasiconvex vector optimization and applications to multi-task learning](https://mlsanity.com/paper/2402.06224/) (2024)

## Abstract (arXiv)

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.