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

ABSTRACT

This paper deals with the Frank–Wolfe algorithm to solve a special class of non-compact constrained optimization problems. The notion of asymptotic cone is one the main concept used to introduce the class of problems considered as well as to establish the well definition of the algorithm. This class of optimization problems, with closed and convex constraint set, are characterized by two conditions on the gradient of the objective function. The first one establishes that the gradient of the objective function is Lipschitz continuous, which is quite usual in the analysis of this algorithm. The second one, which is new in this subject, establishes that the gradient belongs to the interior of dual asymptotic cone of the constraint set. Classical results on asymptotic behaviour and iteration complexity bounds for the sequence generated by Frank–Wolfe algorithm are extended to this new class of problems. Some examples of problems with non-compact constraints and objective functions satisfying the aforementioned conditions are provided.

Keywords:

• Frank–Wolfe method
• constrained optimization problem
• non-compact constraint

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grants 305158/2014-7,302473/2017-3, FAPEG/PRONEM-201710267000532 and CAPES. The second author was supported in part by Fundação de Apoio à Pesquisa do Distrito Federal (FAP-DF) by the grant 0193.001695/2017,PDE 05/2018. This research was carried out, in part, during (the state of alert in the Catalonia) a visit, of the second author to the Centro de Recerca Matemática (CRM), in the framework of the Research in pairs call in 2020. The CRM is a paradise for research, the author appreciates the hospitality and all the support received from CRM.