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.     

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.     

Generalized Constraint Qualifications and Optimality Conditions for Set-Valued Optimization Problems

In this paper we discuss the connections of four generalized constraint qualifications for set-valued vector optimization problems with constraints. Then some K-T type necessary and sufficient optimality conditions are derived, in terms of the contingent epiderivatives.     

Necessary Optimality Conditions for Higher-Order Infinite Horizon Variational Problems on Time Scales

We obtain Euler?CLagrange and transversality optimality conditions for higher-order infinite horizon variational problems on a time scale. The new necessary optimality conditions improve the classical results both in the continuous and discrete settings: Our results seem new and interesting even in the particular cases when the time scale is the set of real numbers or the set of integers.     

Necessary Optimality Conditions for Bilevel Optimization Problems Using Convexificators

In this work, we use a notion of convexificator (Jeyakumar, V. and Luc, D.T. (1999), Journal of Optimization Theory and Applicatons, 101, 599–621.) to establish necessary optimality conditions for bilevel optimization problems. For this end, we introduce an appropriate regularity condition to help us discern the Lagrange–Kuhn–Tucker multipliers.     

集值优化问题的Benson真有效解的广义最优性条件

引进了关于集值映射的(1,α)-阶Clarke导数,(1,α)-阶邻接导数,(1,α)-阶伴随导数概念;应用它们导出了具Slater约束规格的集值优化问题的Benson真有效解的广义导数型Kuhn-Tucker最优性条件。     

Necessary Optimality Conditions for Mathematical Programs with Second-Order Cone Complementarity Constraints

This paper considers the mathematical program with second-order cone complementarity constrains (MPSOCC). As a generalization of the developed mathematical program with complementarity constrains (MPCC), MPSOCC has many applications in practice. Motivated by the MPCC theory, several stationarity concepts, which include the Clarke-type, Mordukhovich-type, and strong stationarities, are presented in this paper. It is further shown that a local minimizer of MPSOCC must be stationary in some sense under suitable conditions. This indicates that these stationarity concepts are reasonable in theory.     

Benson真有效解意义下集值优化问题的最优性条件

引进了集值映射关于锥的(1,α)-阶Clarke切导数,(1,α)-阶Adjacent切导数,(1,α)-阶Contingent切导数概念;应用它们导出了具Slater约束规格的集值优化问题的Benson真有效解的广义Kuhn-Tucker最优性条件．     

Necessary Optimality Conditions for Interval Optimization Problems with Functional and Abstract Constraints

Journal of Optimization Theory and Applications - This work addresses interval optimization problems in which the objective function is interval-valued while the constraints are given in functional...     

Multifunction Optimization Problems Involving Parameters: Necessary Optimality Conditions

We prove the Fritz John and Kuhn-Tucker necessary optimality conditions for vector optimization problems involving multifunctions and parameters under relaxed assumptions.     

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.     

Necessary Optimality Conditions and New Optimization Methods for Cubic Polynomial Optimization Problems with Mixed Variables

Multivariate cubic polynomial optimization problems, as a special case of the general polynomial optimization, have a lot of practical applications in real world. In this paper, some necessary local optimality conditions and some necessary global optimality conditions for cubic polynomial optimization problems with mixed variables are established. Then some local optimization methods, including weakly local optimization methods for general problems with mixed variables and strongly local optimization methods for cubic polynomial optimization problems with mixed variables, are proposed by exploiting these necessary local optimality conditions and necessary global optimality conditions. A global optimization method is proposed for cubic polynomial optimization problems by combining these local optimization methods together with some auxiliary functions. Some numerical examples are also given to illustrate that these approaches are very efficient.     

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.     

Optimality Conditions of Generalized Subconvexlike Set-Valued Optimization Problems Based on the Quasi-Relative Interior

In this paper, firstly, a new generalized subconvexlike set-valued map based on the quasi-relative interior is introduced. Secondly, by a separation theorem involving the quasi-relative interior, some separation properties are obtained. Finally, some optimality conditions are established. Our results improve some results in the literature.     

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. This work was supported by the National Natural Science Foundation of China (10671135), the Specialized Research Fund for the Doctoral Program of Higher Education (20060610005) and the National Natural Science Foundation of Sichuan Province (07ZA123). The authors thank Professor P.M. Pardalos and the referees for comments and suggestions.     

集值优化问题严有效解的高阶标量化定理

讨论了集值优化问题严有效解的高阶导数型标量化定理.首先得到了集值优化问题严有效解的一个高阶导数型必要性条件,其次获得了集值优化问题严有效解的标量化必要性条件和充分性条件.     

First-Order Necessary Optimality Conditions for General Bilevel Programming Problems

In Ref. 1, bilevel programming problems have been investigated using an equivalent formulation by use of the optimal value function of the lower level problem. In this comment, it is shown that Ref. 1 contains two incorrect results: in Proposition 2.1, upper semicontinuity instead of lower semicontinuity has to be used for guaranteeing existence of optimal solutions; in Theorem 5.1, the assumption that the abnormal part of the directional derivative of the optimal value function reduces to zero has to be replaced by the demand that a nonzero abnormal Lagrange multiplier does not exist.