The projected gradient direction in vector optimization: continuity and dual characterization

This technical note proves that, for a smooth vector optimization problem on a closed convex feasible set ordered by a pointed cone, the projected gradient direction depends continuously on the decision variable. Our argument is based on a simple and direct proof via a fixed-domain reformulation of the subproblem. We then give a necessary and sufficient dual characterization of this direction and show that its associated set-valued dual variable mapping is outer semicontinuous.