Conditional gradient method for vector optimization

TL;DR

This paper introduces a conditional gradient method for constrained vector optimization problems, extending existing multiobjective optimization techniques and providing convergence guarantees under various step size strategies.

Contribution

It proposes a novel conditional gradient algorithm for vector optimization with convergence analysis, including new conditions for adaptive step size strategies.

Findings

Proves stationarity of accumulation points with Armijo and nonmonotone steps.

Establishes convergence to weakly efficient solutions under convexity.

Generalizes descent lemma for vector optimization.

Abstract

In this paper, we propose a conditional gradient method for solving constrained vector optimization problems with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. When the partial order under consideration is the one induced by the non-negative orthant, we regain the method for multiobjective optimization recently proposed by Assun\c{c}\~{a}o et al. (Comput Optim Appl 78(3):741--768, 2021). In our method, the construction of auxiliary subproblem is based on the well-known oriented distance function. Three different types of step size strategies (Armijio, adaptative and nonmonotone) are considered. Without any assumptions, we prove that stationarity of accumulation points of the sequences produced by the proposed method equipped with the Armijio or the nonmonotone step size rule. To obtain the convergence result of the method with the adaptative step size strategy, we introduce an useful cone convexity condition which allows to circumvent the intricate question of the Lipschitz continuity of Jocabian for the objective function. This condition helps us generalize the classical descent lemma to the vector optimization case. Under suitable convexity assumptions for the objective function, it is proved that all accumulation points of any generated sequences obtained by our method are weakly efficient solutions.