Higher-order optimality conditions for proper efficiency in nonsmooth vector optimization using radial sets and radial derivatives

We establish both necessary and sufficient optimality conditions of higher orders for various kinds of proper solutions to nonsmooth vector optimization in terms of higher-order radial sets and radial derivatives. These conditions are for global solutions and do not require continuity and convexity assumptions. Examples are provided to show advantages of the results over existing ones in a number of cases.     

Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality

In this paper, we introduce upper and lower Studniarski derivatives of set-valued maps. By virtue of these derivatives, higher-order necessary and sufficient optimality conditions are obtained for several kinds of minimizers of a set-valued optimization problem. Then, applications to duality are given. Some remarks on several existent results and examples are provided to illustrate our results.     

Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization

We propose notions of higher-order outer and inner radial derivatives of set-valued maps and obtain main calculus rules. Some direct applications of these rules in proving optimality conditions for particular optimization problems are provided. Then we establish higher-order optimality necessary conditions and sufficient ones for a general set-valued vector optimization problem with inequality constraints. A number of examples illustrate both the calculus rules and the optimality conditions. In particular, they explain some advantages of our results over earlier existing ones and why we need higher-order radial derivatives.     

Higher-order optimality conditions for set-valued optimization with ordering cones having empty interior using variational sets

In this paper, we first establish chain rules and sum rules for variational sets of type 2. For their applications, optimality conditions of two particular optimization problems are discussed. Then, we obtain higher-order optimality conditions for proper Henig solutions of a set-valued optimization problem in terms of variational sets of type 2 when ordering cones have empty interior.     

Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization

In this paper, generalized higher-order contingent (adjacent) derivatives of set-valued maps are introduced and some of their properties are discussed. Under no any convexity assumptions, necessary and sufficient optimality conditions are obtained for weakly efficient solutions of set-valued optimization problems by employing the generalized higher-order derivatives.     

Higher-order optimality conditions for strict and weak efficient solutions in set-valued optimization

In this paper, we introduce a notion of higher-order Studniarski epiderivative of a set-valued map and study its properties. Then, we discuss their applications to optimality conditions in set-valued optimization. Higher-order optimality conditions for strict and weak efficient solutions of a constrained set-valued optimization problem are established. Some remarks on the existing results in the literature are given from our results.     

Second-order optimality conditions in set-valued optimization via asymptotic derivatives

In this article we give new second-order optimality conditions in set-valued optimization. We use the second-order asymptotic tangent cones to define second-order asymptotic derivatives and employ them to give the optimality conditions. We extend the well-known Dubovitskii–Milutin approach to set-valued optimization to express the optimality conditions given as an empty intersection of certain cones in the objective space. We also use some duality arguments to give new multiplier rules. By following the more commonly adopted direct approach, we also give optimality conditions in terms of a disjunction of certain cones in the image space. Several particular cases are discussed.     

First-order optimality conditions in set-valued optimization

A a set-valued optimization problem min _C F(x), x ∈X ₀, is considered, where X ₀ ⊂ X, X and Y are normed spaces, F: X ₀ ⊂ Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x ⁰,y ⁰), y ⁰ ∈F(x ⁰), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini directional derivative first order necessary conditions and sufficient conditions a pair (x ⁰, y ⁰) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done.     

Necessary optimality conditions for weak sharp minima in set-valued optimization

The aim of the present paper is to get necessary optimality conditions for a general kind of sharp efficiency for set-valued mappings in infinite dimensional framework. The efficiency is taken with respect to a closed convex cone and as the basis of our conditions we use the Mordukhovich generalized differentiation. We have divided our work into two main parts concerning, on the one hand, the case of a solid ordering cone and, on the other hand, the general case without additional assumptions on the cone. In both situations, we derive some scalarization procedures in order to get the main results in terms of the Mordukhovich coderivative, but in the general case we also carryout a reduction of the sharp efficiency to the classical Pareto efficiency which, in addition with a new calculus rule for Fréchet coderivative of a difference between two maps, allows us to obtain some results in Fréchet form.     

Second-order optimality conditions for cone-subarcwise connected set-valued optimization problems

The concept of a cone subarcwise connected set-valued map is introduced. Several examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then, by virtue of the generalized second-order contingent epiderivative, second-order necessary optimality conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problems. When objective function is cone subarcwise connected, a second-order sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problems.     

On optimality conditions for quasi-relative efficient solutions in set-valued optimization

In the paper, we establish necessary and sufficient optimality conditions for quasi-relative efficient solutions of a constrained set-valued optimization problem using the Lagrange multipliers. Many examples are given to show that our results and their applications are more advantageous than some existing ones in the literature.     

First and second order optimality conditions for set-valued optimization problems

In this paper we study first and second order necessary and sufficient optimality conditions for optimization problems involving set-valued maps and we derive some known results in a more general framework.     

Higher-order pointwise optimality conditions

Necessary higher-order optimality conditions are given for nonlinear smooth nonstationary control systems, linear with respect to control and defined on a smooth finite-dimensional manifold; the admissible values of the control belong to a closed convex polyhedron, while the initial and final times and the initial and final points of the trajectory are not fixed. Unlike the previously known necessary optimality conditions, the paper does not require that the passage from one condition, of a given sequence of necessary conditions, to another one should be performed only when the preceding condition degenerates on a time interval of nonzero length.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 39, pp. 118–177, 1991.     

Second-order necessary optimality conditions for optimization problems involving set-valued maps

Second-order necessary optimality conditions are established under a regularity assumption for a problem of minimizing a functiong over the solution set of an inclusion system 0 F(x), x M, whereF is a set-valued map between finite-dimensional spaces andM is a given subset. The proof of the main result of the paper is based on the theory of infinite systems of linear inequalities.     

Higher order optimality conditions for Henig efficient solutions in set-valued optimization

In this paper, higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set-valued maps are introduced. Necessary and sufficient conditions for Henig efficient solutions to a constrained set-valued optimization problem are given by employing the higher order generalized epiderivatives.     

Set-relations and optimality conditions in set-valued maps

Problems in set-valued optimization can be solved via set-optimization. In this paper properties of set-optimization are investigated. Conditions for existence of solutions are established. Directional derivatives are studied for set-valued mappings. Necessary and sufficient conditions in the existence of solutions are showed with directional derivatives.     

Necessary optimality conditions for set-valued optimization problems via the extremal principle

The paper concerns first-order necessary optimality conditions for set-valued optimization problems. Based on the extremal principle developed by Mordukhovich [21], we derive fuzzy/approximate necessary optimality conditions. An example that illustrates the usefulness of our results is given.