Generalized Proximal Method for Efficient Solutions in Vector Optimization

Abstract

This article is devoted to developing the generalized proximal algorithm of finding efficient solutions to the vector optimization problem for a mapping from a uniformly convex and uniformly smooth Banach space to a real Banach space with respect to the partial order induced by a pointed closed convex cone. In contrast to most published literature on this subject, our algorithm does not depend on the nonemptiness of ordering cone of the space under consideration and deals with finding efficient solutions of the vector optimization problem in question. We prove that under some suitable conditions the sequence generated by our method weakly converges to an efficient solution of this problem.

Approximate Solutions of Common Fixed-Point Problems

“…Proximal point method is an important tool in solving optimization problems [4,42,44,56,59,61,68,69,78,88]. It is also used for solving variational inequalities with monotone operators [2, 8, 11, 17-21, 25, 57, 60, 62-64, 79, 82, 83, 91, 92] which is an important topic of nonlinear analysis and optimization [9,14,28,29,36,51,55,58,65,66,80,81,86,87,90].…”

A proximal point method for difference of convex functions in multi-objective optimization with application to group dynamic problems

“…This descent property plays an important role for a large class of application problems; see for instance Bento and Soubeyran [7]. Other authors have considered variants of the proximal method proposed by Bonnel et al [10] using the improving constraint F (x k+1 ) C F (x k ), for some ordering cone C; for instance Apolinário et al [1], Bento et al [5], Ceng and Yao [13], Ceng et al [14], Choung et al [15], Villacorta and Oliveira [66]. However, the constrained set Ω k is convex in all these works.…”

“…Then, we will provide a multi-objective DC algorithm which converges to a Pareto critical point, with a very important added requirement: the algorithm must follow a cooperative improving dynamic x k+1 ∈ Ω(x k ). In this behavioral setting, the papers of Bento et al [5], Bonnel et al [10] and Choung et al [15] make "as if" they consider this case of a cooperative improving dynamic. However, the aforementioned works use convexity of the improving sets Ω(x k ), for all k ∈ N. A motivation, in a dynamic context, to consider the constrained set Ω(x k ) is given in Bento et al [5].…”

The Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization with Application to the Compromise Problem

“…Other authors have proposed variants of the algorithm considered by Bonnel, Iusem, and Svaiter [7] for convex vector or multiobjective problems; see, for instance, Ceng and Yao [11], Ceng, Mordukhovich, and Yao [12], Choung, Mordukhovich, and Yao [13], Gregório and Oliveira [31], and Villacorta and Oliveira [52]. Recently, the R m + -quasi-convex case was discussed in Bento, Cruz Neto, and Soubeyran [4] and Apolinário, Papa Quiroz, and Oliveira [1]; see the definition of R m + -quasi-convexity on section 2.…”