Approximate generalized proximal-type method for convex vector optimization problem in Banach spaces

In this paper, we consider a convex vector optimization problem of finding weak Pareto optimal solutions for an extended vector-valued map from a uniformly convex and uniformly smooth Banach space to a real Banach space, with respect to the partial order induced by a closed, convex and pointed cone with a nonempty interior. We propose an inexact vector-valued proximal-type point algorithm based on a Lyapunov functional when the iterates are computed approximately and prove the sequence generated by the algorithm weakly converges to a weak Pareto optimal solution of the vector optimization problem under some mild conditions. Our results improve and generalize some known results.