  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.
