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.

摘要

本文讨论了Frank-Wolfe算法，以解决一类特殊的非紧致约束优化问题。渐近锥的概念是用于介绍所考虑问题的类别以及建立算法的良好定义的主要概念之一。这类具有封闭和凸约束集的优化问题的特征在于目标函数梯度的两个条件。第一个建立目标函数的梯度是Lipschitz连续的，这在该算法的分析中很常见。第二个在本主题中是新的，它确定梯度属于约束集的对偶渐近锥的内部。关于Frank-Wolfe算法生成的序列的渐近行为和迭代复杂度界限的经典结果被扩展到这一类新问题。