Solutions Of Vector Optimization Problems Research Articles

Some properties of approximate solutions for vector optimization problem with set-valued functions

In this paper, we study the approximate solutions for vector optimization problem with set-valued functions. The scalar characterization is derived without imposing any convexity assumption on the objective functions. The relationships between approximate solutions and weak efficient solutions are discussed. In particular, we prove the connectedness of the set of approximate solutions under the condition that the objective functions are quasiconvex set-valued functions.

Optimality Conditions for Vector Optimization Problems

In this paper, some necessary and sufficient optimality conditions for the weakly efficient solutions of vector optimization problems (VOP) with finite equality and inequality constraints are shown by using two kinds of constraints qualifications in terms of the MP subdifferential due to Ye. A partial calmness and a penalized problem for the (VOP) are introduced and then the equivalence between the weakly efficient solution of the (VOP) and the local minimum solution of its penalized problem is proved under the assumption of partial calmness.

Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets

In this paper, the nonemptiness and compactness of solution sets for Stampacchia vector variational-like inequalities (for short, SVVLIs) and Minty vector variational-like inequalities (for short, MVVLIs) with generalized bifunctions defined on nonconvex sets are investigated by introducing the concepts of generalized weak cone-pseudomonotonicity and generalized (proper) cone-suboddness. Moreover, some equivalent relations between a solution of SVVLIs and MVVLIs, and a generalized weakly efficient solution of vector optimization problems (for short, VOPs) are established under the assumptions of generalized pseudoconvexity and generalized invexity in the sense of Clarke generalized directional derivative. These results extend and improve the corresponding results of others.

Application of the set-theoretic approach to accounting of uncertainties in the solution of vector optimization problems

The problem of optimization of a complex system by the vector criterion is considered in terms of the set theory. The suggested approach to its solution is based on the formal description of a set of the weight factors of partial indices of effectiveness by the system of inequalities proceeding from the nonstrict and incomplete system of preferences of the decision-maker (DM), which enables us to reduce the calculation of values of both the integral and the guaranteeing criterion of optimality to the solution of a simple geometric problem

On Minty vector variational-like inequality

This paper is devoted to the study of relationships between solutions of Minty vector variational-like inequalities (MVVLI) and solutions of vector optimization problems (VOP), as well as some relations between solutions of MVVLI and Stampacchia vector variational-like inequalities (SVVLI). Moreover, some relations between MVVLI and perturbed vector variational-like inequalities are established. Our results generalize some known results.

Merit functions in vector optimization

We study the weak domination property and weakly efficient solutions in vector optimization problems. In particular scalarization of these problems is obtained by virtue of some suitable merit functions. Some natural conditions to ensure the existence of error bounds for merit functions are also given.

Stability Results for Efficient Solutions of Vector Optimization Problems

Using the additive weight method of vector optimization problems and the method of essential solutions, we study some continuity properties of the mapping which associates the set of efficient solutions S(f) to the objective function f. To understand such properties, the key point is to consider the stability of additive weight solutions and the relationship between efficient solutions and additive weight solutions.

Optimality Conditions for Metrically Consistent Approximate Solutions in Vector Optimization

In this paper, approximate solutions of vector optimization problems are analyzed via a metrically consistent ?-efficient concept. Several properties of the ?-efficient set are studied. By scalarization, necessary and sufficient conditions for approximate solutions of convex and nonconvex vector optimization problems are provided; a characterization is obtained via generalized Chebyshev norms, attaining the same precision in the vector problem as in the scalarization.

THEORETICAL BASIS FOR THE DETERMINATION OF RATIONAL PARAMETERS OF COMPLEX MECHANICAL SYSTEMS

The mathematical description of choosing the parameters of mechanical systems in general form and with eking account of the features of rail vehicles is presented. The examples of solution of vector optimization problems according to several indices are considered.

On comparison of approximate solutions in vector optimization problems

The problem of comparison of approximations (approximate solutions to a vector optimization problem) obtained using different numerical methods is considered. In the absence of a priori information about the set of weakly efficient vectors, a scalar function is introduced that enables pair-wise comparison of approximations and establishes a binary preference relation according to which the approximations close (in the sense of the Hausdorff distance) to the set containing all possible efficient vectors are preferable to other approximations.

On Approximate Solutions in Vector Optimization Problems Via Scalarization

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 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.

Duality and saddle-points for convex-like vector optimization problems on real linear spaces

Usually, finite dimensional linear spaces, locally convex topological linear spaces or normed spaces are the framework for vector and multiojective optimization problems. Likewise, several generalizations of convexity are used in order to obtain new results. In this paper we show several Lagrangian type duality theorems and saddle-points theorems. From these, we obtain some characterizations of several efficient solutions of vector optimization problems (VOP), such as weak and proper efficient solutions in Benson’s sense. These theorems are generalizations of preceding results in two ways. Firstly, because we consider real linear spaces without any particular topology, and secondly because we work with a recently appeared convexlike type of convexity. This new type, designated GVCL in this paper, is based on a new algebraic closure which we named vector closure.

On essential sets and essential components of efficient solutions for vector optimization problems

In this paper, we first introduce the notions of an essential set and an essential component of the set of efficient solutions for continuous vector optimizations on a nonempty compact subset of a metric space. Then we show that for each of these vector optimizations, each set of all efficient solutions corresponding to the same optimal values is essential. Basing on this result, we give full characterizations of an essential point, an essential set and an essential component, respectively. As an application, we prove that for continuous quasiconvex vector optimization problems on a nonempty compact subset of a metric vector space, each component of the set of efficient solutions is essential even though the efficient solution set is not connected.

A Nonlinear Scalarization Function and Generalized Quasi-vector Equilibrium Problems

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. In this paper we introduce a nonlinear scalarization function for a variable domination structure. Several important properties, such as subadditiveness and continuity, of this nonlinear scalarization function are established. This nonlinear scalarization function is applied to study the existence of solutions for generalized quasi-vector equilibrium problems.

Ideal, weakly efficient solutions for vector optimization problems

We establish existence results for finite dimensional vector minimization problems allowing the solution set to be unbounded. The solutions to be referred are ideal(strong)/weakly efficient(weakly Pareto). Also several necessary and/or sufficient conditions for the solution set to be non-empty and compact are established. Moreover, some characterizations of the non-emptiness (boundedness) of the (convex) solution set in case the solutions are searched in a subset of the real line, are also given. However, the solution set fails to be convex in general. In addition, special attention is addressed when the underlying cone is the non-negative orthant and when the semi-strict quasiconvexity of each component of the vector-valued function is assumed. Our approach is based on the asymptotic description of the functions and sets. Some examples illustrating such an approach are also exhibited.

On Relations Between Vector Optimization Problems and Vector Variational Inequalities

We obtain equivalences between weak Pareto solutions of vector optimization problems and solutions of vector variational inequalities involving generalized directional derivatives.

Generalized properly efficient solutions of vector optimization problems

Generalized properly efficient solutions of a vector optimization problem (VP) are defined in terms of various tangent cones and a generalized directional derivative. We study their basic properties and relationships and show that under certain conditions, a generalized properly efficient solution of (VP), defined by the adjacent cone, is a generalized Kuhn-Tucker properly efficient solution of (VP). Furthermore, using subgradients defined by closed convex tangent cones, we give a necessary optimality condition for a generalized properly efficient solution of (VP) defined by the adjacent cone.

A Result on Vector Variational Inequalities with Polyhedral Constraint Sets

In this note, by using some well-known results on properly efficient solutions of vector optimization problems, we show that the Pareto solution set of a vector variational inequality with a polyhedral constraint set can be expressed as the union of the solution sets of a family of (scalar) variational inequalities.

∈-Weak Minimal Solutions of Vector Optimization Problems with Set-Valued Maps

In this paper, we consider cone-subconvexlike vector optimization problems with set-valued maps in general spaces and derive scalarization results, ∈-saddle point theorems, and ∈-duality assertions using ∈-Lagrangian multipliers.