Conditional gradient method for vector optimization

Abstract

In this paper, we propose an extension of the classical Frank-Wolfe method for solving constrained vector optimization problems with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. In the proposed method, the construction of auxiliary subproblem is based on the well-known oriented distance function. Two types of stepsize strategies including Armijio line search and adaptive stepsize are used. It is shown that every accumulation point of the generated sequences satisfies the first-order necessary optimality condition. Moreover, under suitable convexity assumptions for the objective function, it is proved that all accumulation points of any generated sequences are weakly efficient points. We finally apply the proposed algorithms to a portfolio optimization problem under bicriteria considerations.

Convergence rates analysis of Interior Bregman Gradient Method for Vector Optimization Problems

“…, the (L, C, e)-smoothness reduces to condition (A) in [31]. If C = R + , the (L, C, e)-smoothness and (µ, C, e)-strong convexity correspond to relative L-smoothness and µ-strong convexity in [21], respectively.…”