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.     

Second-Order Optimality Conditions for Strict Efficiency of Constrained Set-Valued Optimization

In this paper, we propose several second-order derivatives for set-valued maps and discuss their properties. By using these derivatives, we obtain second-order necessary optimality conditions for strict efficiency of a set-valued optimization problem with inclusion constraints in real normed spaces. We also establish second-order sufficient optimality conditions for strict efficiency of the set-valued optimization problem in finite-dimensional normed spaces. As applications, we investigate second-order sufficient and necessary optimality conditions for a strict local efficient solution of order two of a nonsmooth vector optimization problem with an abstract set and a functional constraint.     

Second-Order Optimality Conditions in Set Optimization

In this paper, we propose second-order epiderivatives for set-valued maps. By using these concepts, second-order necessary optimality conditions and a sufficient optimality condition are given in set optimization. These conditions extend some known results in optimization.The authors are grateful to the referees for careful reading and helpful remarks.     

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.     

Higher-order generalized Studniarski epiderivative and its applications in set-valued optimization

In the paper, we introduce the higher-order generalized Studniarski epiderivative of set-valued maps. Via this concept, some results on optimality conditions and duality for set-valued optimization problems are established.     

Higher-Order Optimality Conditions for Set-Valued Optimization

This paper deals with higher-order optimality conditions of set-valued optimization problems. By virtue of the higher-order derivatives introduced in (Aubin and Frankowska, Set-Valued Analysis, Birkhäuser, Boston, [1990]) higher-order necessary and sufficient optimality conditions are obtained for a set-valued optimization problem whose constraint condition is determined by a fixed set. Higher-order Fritz John type necessary and sufficient optimality conditions are also obtained for a set-valued optimization problem whose constraint condition is determined by a set-valued map.     

方向导数和广义锥-预不变凸集值优化问题

讨论拓扑向量空间中无约束集值优化问题的最优性条件问题.利用集值映射的Dini方向导数,在广义锥-预不变凸性条件下,建立了集值优化问题关于弱极小元和强极小元的最优性充分必要条件.     

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.     

Optimality conditions in set optimization employing higher-order radial derivatives

There are two approaches of defining the solutions of a set-valued optimization problem:vector criterion and set criterion.This note is devoted to higher-order optimality conditions using both criteria of solutions for a constrained set-valued optimization problem in terms of higher-order radial derivatives.In the case of vector criterion,some optimality conditions are derived for isolated (weak) minimizers.With set criterion,necessary and sufficient optimality conditions are established for minimal solutions relative to lower set-order relation.     

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.     

集值优化问题Benson真有效解的高阶Fritz John型最优性条件

本文讨论的是集值优化问题Benson真有效解的高阶Fritz John型最优性条件,利用Aubin和Fraukowska引入的高阶切集和凸集分离定理,在锥-似凸映射的假设条件下,获得了带广义不等式约束的集值优化问题Benson真有效解的高阶Fritz John型必要和充分性条件.     

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 Karush–Kuhn–Tucker optimality conditions for set-valued optimization with nonsolid ordering cones

In this paper, we consider higher-order Karush–Kuhn–Tucker optimality conditions in terms of radial derivatives for set-valued optimization with nonsolid ordering cones. First, we develop sum rules and chain rules in the form of equality for radial derivatives. Then, we investigate set-valued optimization including mixed constraints with both ordering cones in the objective and constraint spaces having possibly empty interior. We obtain necessary conditions for quasi-relative efficient solutions and sufficient conditions for Pareto efficient solutions. For the special case of weak efficient solutions, we receive even necessary and sufficient conditions. Our results are new or improve recent existing ones in the literature.     

Generalized Higher-Order Optimality Conditions for Set-Valued Optimization under Henig Efficiency

In this paper, generalized mth-order contingent epiderivative and generalized mth-order epiderivative of set-valued maps are introduced, respectively. By virtue of the generalized mth-order epiderivatives, generalized necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Generalized Kuhn–Tucker type necessary and sufficient optimality conditions are also obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.     

On Optimization Problems with Set-Valued Objective Maps: Existence and Optimality

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets of n-dimensional Euclidean space. Second, by using these quasi orderings, we define the concepts of lower semi-continuity for set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and we give some conditions under which these optimal solutions exist to the problems and give necessary and sufficient conditions for optimality.     

集值优化问题的高阶最优性条件(借助Studniarski导数)(英文)

本文研究的是约束集值优化问题的高价最优性条件.首先通过借助集值映射的Stud-niarski导数和严格局部有效性,讨论了集值优化问题的高阶必要条件和充分条件.对于充分条件,初始空间必须是有限维的.其次在初始空间和目标空间是有限维的以及集值映射是m阶稳定的条件下,也得到了此约束集值优化问题的高阶最优性条件.     

Optimality conditions for set-valued maps with set optimization

Problems in set-valued optimization can be solved via set optimization. In this paper optimality conditions are studied for set-valued maps with set optimization. Optimality requirements are established for continuous selections using directional derivatives. Necessary and sufficient conditions for the existence of solutions are shown for set-valued maps under generalized convexity assumptions and with the notion of the contingent derivative.     

On super efficiency in set-valued optimization

The set-valued optimization problem with constraints is considered in the sense of super efficiency in locally convex linear topological spaces. Under the assumption of iccone-convexlikeness, by applying the seperation theorem, Kuhn-Tucker＇s, Lagrange＇s and saddle points optimality conditions, the necessary conditions are obtained for the set-valued optimization problem to attain its super efficient solutions. Also, the sufficient conditions for Kuhn-Tucker＇s, Lagrange＇s and saddle points optimality conditions are derived.     

Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives

We propose higher-order radial sets and corresponding derivatives of a set-valued map and prove calculus rules for sums and compositions, which are followed by direct applications in discussing optimality conditions for several particular optimization problems. Our main results are both necessary and sufficient higher-order conditions for weak efficiency in a general set-valued vector optimization problem without any convexity assumptions. Many examples are provided to explain advantages of our results over a number of existing ones in the literature.