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

Generalized Proximal Method for Efficient Solutions in Vector Optimization

“…The new algorithm does not impose any interiority assumptions on the ordering cone under consideration while using the standing assumptions formulated in section 2 with some r ∈ (0, 1] from (2.7) as well as use r -proper efficient solutions to the subproblems instead of weak efficient ones as in [8].…”

“…Among the major modifications implemented below, we mention the construction and justification of the so-called r-proper efficient solutions [9] to the subproblems instead of weak efficient ones as in [8] and the previous developments.…”

“…In this article, we will modify the algorithm in [8] to avoid the restrictive interiority assumption in such a way that the constructed iterative sequence weakly converges to an efficient solution of the initial vector optimization problem (1.3). Among the major modifications implemented below, we mention the construction and justification of the so-called r-proper efficient solutions [9] to the subproblems instead of weak efficient ones as in [8] and the previous developments.…”

“…It is worth noticing that the authors in [7,8] introduced and studied the so-called proximal-type and approximate generalized proximal-type (or generalized proximal) methods of finding weak efficient solutions to the convex vector optimization problem for a mapping from a uniformly convex and uniformly smooth Banach space to a real Banach space with the ordering cone C having a nonempty interior. Under some conditions, they proved that any sequence generated by their methods weakly converges to a weak efficient solution of the problem.…”

Conditional gradient method for vector optimization

“…In addition, by virtue of the general order cone C, vector versions of the proximal point method [18], the nonmonotone gradient algorithm [19] and the Hager-Zhang conjugate gradient method [20] are introduced to solve vector optimization problems. In infinite-dimensional settings, there are also several methods for solving vector optimization problems (see [21][22][23][24][25][26][27] and references therein). For example, Chuong and Yao [25] presented exact and inexact steepest descent methods of vector optimization problems for a map from a finite dimensional Hilbert space to a Banach space, which generalizes the works in [4].…”