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A Unified Approach and Optimality Conditions for Approximate Solutions of Vector Optimization Problems

This paper deals with approximate (V-efficient) solutions of vector optimization problems. We introduce a new V-efficiency concept which extends and unifies different approximate solution notions introduced in the literature. We obtain necessary and sufficient conditions via nonlinear scalarization, which allow us to study this new class of approximate solutions in a general framework, since any convexity hypothesis is required. Several examples are proposed to show the concepts introduced and the results attained.

Second order necessary conditions in set constrained differentiable vector optimization

We state second order necessary optimality conditions for a vector optimization problem with an arbitrary feasible set and an order in the final space given by a pointed convex cone with nonempty interior. We establish, in finite-dimensional spaces, second order optimality conditions in dual form by means of Lagrange multipliers rules when the feasible set is defined by a function constrained to a set with convex tangent cone. To pass from general conditions to Lagrange multipliers rules, a generalized Motzkin alternative theorem is provided. All the involved functions are assumed to be twice Fréchet differentiable.

A Set-Valued Ekeland's Variational Principle in Vector Optimization

This paper deals with Ekelands variational principle for vector optimization problems. By using a set-valued metric, a set-valued perturbed map, and a cone-boundedness concept based on scalarization, we introduce an original approach to extending the well-known scalar Ekelands principle to vector-valued maps. As a consequence of this approach, we obtain an Ekelands variational principle that does not depend on any approximate efficiency notion. This result is related to other Ekelands principles proved in the literature, and the finite-dimensional case is developed via an

On Approximate Solutions in Vector Optimization Problems Via Scalarization

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On Approximate Efficiency in Multiobjective Programming

-efficiency notion that we introduced in [Math. Methods Oper. Res., 64 (2006), pp. 165-185; SIAM J. Optim., 17 (2006), pp. 688-710].

Improvement sets and vector optimization

This work deals with approximate solutions in vector optimization problems. These solutions frequently appear when an iterative algorithm is used to solve a vector optimization problem. We consider a concept of approximate efficiency introduced by Kutateladze and widely used in the literature to study this kind of solutions. Necessary and sufficient conditions for Kutateladze’s approximate solutions are given through scalarization, in such a way that these points are approximate solutions for a scalar optimization problem. Necessary conditions are obtained by using gauge functionals while monotone functionals are considered to attain sufficient conditions. Two properties are then introduced to describe the idea of parametric representation of the approximate efficient set. Finally, through scalarization, characterizations and parametric representations for the set of approximate solutions in convex and nonconvex vector optimization problems are proved and the obtained results are applied to Pareto problems.

Multiplier Rules and Saddle-Point Theorems for Helbig's Approximate Solutions in Convex Pareto Problems

AbstractThis paper is focused on approximate (

Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness

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