Optimality conditions for proper efficient solutions of vector set-valued optimization

Based on the concept of an epiderivative for a set-valued map introduced in J. Nanchang Univ. 25 (2001) 122-130, in this paper, we present a few necessary and sufficient conditions for a Henig efficient solution, a globally proper efficient solution, a positive properly efficient solution, an f-efficient solution and a strongly efficient solution, respectively, to a vector set-valued optimization problem with constraints.     

Optimality conditions for vector optimization problems with variable ordering structures

Our main concern in this article are concepts of nondominatedness w.r.t. a variable ordering structure introduced by Yu [P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974), pp. 319–377]. Our studies are motivated by some recent applications e.g. in medical image registration. Restricting ourselves to the case when the values of a cone-valued map defining the ordering structure are Bishop–Phelps cones, we obtain for the first time scalarizing functionals for nondominated elements, Fermat rule, Lagrange multiplier rule and duality results for a single- or set-valued vector optimization problem with a variable ordering structure.     

Abstract cone approximations and generalized differentiability in nonsmooth optimization

In the present paper a connection between cone approximations of sets and generalized differentiability notions will be given. Using both conceptions we present an approach to derive necessary optimality conditions for optimization problems with inequality constraints. Moreover, several constraint qualifications are proposed to get Kuhn-Tucker-type-conditions.     

Necessary optimality conditions in nondifferentiable vector optimization

The purpose of the present paper is to give necessary optimality conditions for weak Pareto minimun, peints of nondifferentiable vector optimization problems vcing generalized definitions of the upper and lower Dini-Hadamard derivatives. We give two different approaches for such definitions, a global one and a componentwise one     

Optimality conditions in generalized geometric programming

Generalizations of the Kuhn-Tucker optimality conditions are given, as are the fundamental theorems having to do with their necessity and sufficiency.This research was sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-73-2516.     

On optimality conditions for multiobjective optimization problems in topological vector space

In this paper, we are concerned with a differentiable multiobjective programming problem in topological vector spaces. An alternative theorem for generalized K subconvexlike mappings is given. This permits the establishment of optimality conditions in this context: several generalized Fritz John conditions, in line to those in Hu and Ling [Y. Hu, C. Ling, The generalized optimality conditions of multiobjective programming problem in topological vector space, J. Math. Anal. Appl. 290 (2004) 363-372] are obtained and, in the presence of the generalized Slater's constraint qualification, the Karush-Kuhn-Tucker necessary optimality conditions.     

Optimality conditions of set-valued optimization problem involving relative algebraic interior in ordered linear spaces

In this article, firstly, a generalized cone subconvexlike set-valued map involving the relative algebraic interior is introduced in ordered linear spaces. Secondly, some properties of a generalized cone subconvexlike set-valued map are investigated. Finally, the optimality conditions of set-valued optimization problem are established.     

A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations

In this paper, we consider a class of mathematical programs governed by second-order cone constrained parameterized generalized equations. We reformulate the necessary optimality conditions as a system of nonsmooth equations under linear independence constraint qualification and the strict complementarity condition. A set of second order sufficient conditions is proposed, which is proved to be sufficient for the second order growth of the stationary point. The smoothing Newton method in [40] is employed to solve the system of nonsmooth equations whose strongly BD-regularity at a solution point is demonstrated under the second order sufficient conditions. Several illustrative examples are provided and discussed.     

Optimality conditions in a step control problem

First-and second-order necessary optimality conditions are obtained for the control of step systems.     

The generalized optimality conditions of multiobjective programming problem in topological vector space

In this study, an alternative theorem for the subconvexlike mapping in topological vector space is established. With this alternative theorem as an aid, the generalized Fritz John conditions and the generalized Kuhn-Tucker conditions in terms of Gâteaux derivatives of multiobjective programming problem in the ordered topological vector space are given.     

Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior

In this paper, we consider the set-valued vector optimization problems with constraint in locally convex spaces. We present the necessary and sufficient conditions for Henig efficient solution pair, globally proper efficient solution pair and super efficient solution pair without the ordering cones having the nonempty interior.     

Efficient solutions in generalized linear vector optimization

This paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized linear vector optimization problems is obtained. We also prove that the efficient solution set of a generalized linear vector optimization problem in a locally convex Hausdorff topological vector space is the union of finitely many generalized polyhedral convex sets and it is connected by line segments.     

Necessary optimality conditions in an abnormal optimization problem with equality constraints

An abnormal minimization problem with equality constraints and a finite-dimensional image is examined. Second-order necessary conditions for this problem are given that strengthen previously known results.     

Optimality conditions for non-smooth semi-infinite programming

This article deals with a class of non-smooth semi-infinite programming (SIP) problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. We introduce several kinds of constraint qualifications for these non-smooth SIP problems and we study the relationships between them. Finally, necessary and sufficient optimality conditions are investigated.     

Lagrangian conditions for vector optimization in Banach spaces

We consider vector optimization problems on Banach spaces without convexity assumptions. Under the assumption that the objective function is locally Lipschitz we derive Lagrangian necessary conditions on the basis of Mordukhovich subdifferential and the approximate subdifferential by Ioffe using a non-convex scalarization scheme. Finally, we apply the results for deriving necessary conditions for weakly efficient solutions of non-convex location problems.     

Benson 真有效意义下向量优化的最优性条件

对向量值映射引入了Clarke切导数,A djacent切导数及Contingent导数的概念,讨论了它们的性质,并用其建立了束向量优化有效解在Benson真有效意义下的Fritz John及Kunh-Tucker条件。     

Optimality Conditions for C 1,1 Constrained Multiobjective Problems

The present paper is concerned with the study of the optimality conditions for constrained multiobjective programming problems in which the data have locally Lipschitz Jacobian maps. Second-order necessary and sufficient conditions for efficient solutions are established in terms of second-order subdifferentials of vector functions.     

Optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps characterized by contingent epiderivative

In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.