A Strongly Convergent Proximal Point Method for Vector Optimization

Abstract

In this paper, we propose and analyze a variant of the proximal point method for obtaining weakly efficient solutions of convex vector optimization problems in real Hilbert spaces, with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. The proposed method is a hybrid scheme that combines proximal point type iterations and projections onto some special halfspaces in order to achieve the strong convergence to a weakly efficient solution. To the best of our knowledge, this is the first time that proximal point type method with strong convergence has been considered in the literature for solving vector/multiobjective optimization problems in infinite dimensional Hilbert spaces.

On the strong convergence of the forward-backward splitting algorithm for non-Lipschitz continuous convex optimization problems

“…In some convergence proofs, although the stepsize is obtained using a linesearch method, a positive lower bound is required for the stepsize, which is almost the same as the Lipschitz continuity condition. The study with respect to strong convergence to solve the minimization problem has already been analysed with other techniques (see [9,10,17,18]).…”