集值优化问题的Benson真有效解的广义导数型最优性条件

引进了集值映射关于锥的Clarke切导数, Adjacent切导数与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真有效解的广义导数型最优性条件．     

集值向量优化的二阶最优性必要条件

在赋范空间中给出了集值映射的二阶切集的概念,利用二阶切集,定义了集值映射的二阶切导数。然后,获得了集值向量优化问题弱极小元的两个二阶最优性必要条件。     

Benson真有效意义下集值优化的广义最优性条件

本文引入了关于集值映射的α-阶Clarke切导数、α-阶邻接切导数及α-阶 伴随切导数的概念,借此建立了约束向量集值优化Benson真有效解导数型的Kuhn- Tucker条件.     

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．     

Benson真有效意义下向量集值优化的广义Fritz John条件

引入了一种有关集值映射的切导数和强、弱*伪凸的概念。借助凸集分离定理及锥分离定理建立了Benson真有效意义下向量集值优化导数型的FritzJohn最优性条件,并对条件的充分性进行了讨论。当特殊到单值映射时这些最优性条件与经典的结果完全吻合。     

α-阶近似锥-弧连通集值优化弱有效元的最优性条件

引进了α-阶近似锥-弧连通集值映射,举例说明了它是锥-弧连通集值映射的真推广.借助Y-切锥引进了广义Y-切上图导数,讨论了它与广义切上图导数的关系.当目标函数为α-阶近似锥-弧连通集值映射时,得到集值优化取得弱有效元的充分和必要条件.     

一种新的二阶次梯度及其在刻画集值优化弱有效元中的应用

引进了一种新的修正的二阶切锥,借助此切锥针对集值映射引进了一种新的二阶切导数,讨论了它的若干性质.利用此修正的二阶切导数,引进了一种新的在弱有效元意义下的二阶次梯度,并建立了它的一个存在性定理.借助此二阶次梯度给出了集值优化弱有效元的一种新刻画.     

关于切锥与切导数的等价命题

给出集值映射导数的若干等价命题,这有助于加深对集值映射的导数概念的理解,并给使用带来方便.     

次预不变凸集值优化导数型最优性条件

引入了集值映射的α-阶锥次预不变凸概念,借助于α-阶相依上导数,建立了锥次预不变凸集值映射的导数型择—性定理,并利用择—性定理获得了集值优化导数型的最优性必要条件.     

向量集值优化超有效解的对偶问题

借助于Contingent切锥和集值映射的上图而引入的有关集值映射的Contingent切导数，对约束集值优化问题的超有效解建立了最优性Kuhn Tucker必要及充分性条件,借此建立了向量集值优化超有效解的Wolfe型和Mond Weir型对偶定理.     

Contingent epiderivatives and set-valued optimization

In this paper we introduce the concept of the contingent epiderivative for a set-valued map which modifies a notion introduced by Aubin [2] as upper contingent derivative. It is shown that this kind of a derivative has important properties and is one possible generalization of directional derivatives in the single-valued convex case. For optimization problems with a set-valued objective function optimality conditions based on the concept of the contingent epiderivative are proved which are necessary and sufficient under suitable assumptions.     

Contingent derivatives of set-valued maps and applications to vector optimization

In this paper we investigate contingent derivatives of set-valued maps and their lower and upper semidifferentiability properties. We provide also some calculus rules for these derivatives in infinite dimensional spaces. The concept of contingent derivatives is then applied to produce several necessary and sufficient conditions for vector optimization problems with set-valued objectives.This paper was written when the author was at the University of Erlangen-Nurnberg under a grant of the Alexander von Humboldt Foundation.On leave from the Institute of Mathematics, Hanoi, Vietnam.     

On Some Concepts of Generalized Differentials

In this paper, we study some concepts of generalized differentials for set-valued maps and introduce some new ones. In particular we first focus on the concept of Generalized Differential Quotients, briefly GDQs. It is shown that minimal GDQs are unique for scalar single-valued functions, then GDQs are compared with contingent and Dini derivatives, finally some other results characterizing GDQs are given. A new definition of generalized differentiation theory is presented, namely weak GDQs that are a modification of GDQs. We clarify the relationships with other concepts of generalized differentiability: Clarke generalized Jacobians, path-integral generalized differentials and Warga derivate containers. Finally, some applications of GDQs end the paper.      

Benson真有效意义下向量优化的灵敏度分析

本文用关于集值映射的Contingent切导数定量地讨论了参数映射G（u）在Ben-son真有效意义下的扰动情况.记W（u）=Pmin[G（u）,S],y^∧∈W（u^∧）,则在某些条件下DW（u^∧,y^∧）（u）?Pmin[DG（u^∧,y^∧）（u）],而在另外一些条件下DW（u^∧,y^∧）（u）?Pmin[DG（u^∧,y^∧）（u）].     

Universal contingent claims in a general market environment and multiplicative measures: Examples and applications

We present and further develop the concept of a universal contingent claim introduced by the author in 1995. This concept provides a unified framework for the analysis of a wide class of financial derivatives.A universal contingent claim describes the time evolution of a contingent payoff. In the simplest case of a European contingent claim, this time evolution is given by a family of nonnegative linear operators, the valuation operators. For more complex contingent claims, the time evolution that is given by the valuation operators can be interrupted by discrete or continuous activation of external influences that are described by, generally speaking, nonlinear operators, the activation operators. For example, Bermudan and American contingent claims represent discretely and continuously activated universal contingent claims with the activation operators being the nonlinear maximum operators.We show that the value of a universal contingent claim is given by a multiplicative measure introduced by the author in 1995. Roughly speaking, a multiplicative measure is an operator-valued (in general, an abstract measure with values in a partial monoid) function on a semiring of sets which is multiplicative on the union of disjoint sets. We also show that the value of a universal contingent claim is determined by a, generally speaking, impulsive semilinear evolution equation.     

Characterizations of Optimal Trajectories for Nonconvex Control Systems under State Constraints

We consider an optimal control problem for a nonconvex control system under state constraints and the associated value function, which in general is not differentiable. We provide some characterizations of optimal trajectories using contingent derivatives. For this aim, we derive a costate satisfying the adjoint equation, the maximum principle, and a transversality condition linked to the superdifferential of the value function.Communicated by F. ZirilliThis paper is dedicated by the author to her children.     

Computation formulas and multiplier rules for graphical derivatives in separable Banach spaces

In this paper, we present new computation formulas for the contingent epiderivative and hypoderivative of a set-valued map taking values in a Banach space with a shrinking Schauder basis. These formulas are established in terms of the Fourier coefficients, and, in particular, in terms of the derivatives of the component maps associated with the Schauder basis. As an application, we obtain multiplier rules for vector optimization problems in terms of the derivatives of the component maps, extending classical results from smooth multiobjective optimization problems.     

Implicit multifunction theorems in complete metric spaces

In this paper, we establish some new characterizations of metric regularity of implicit multifunctions in complete metric spaces by using lower semicontinuous envelopes of the distance functions to set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the perturbation stability of implicit multifunctions.     

FILTERED-PROJECTIVE AND SEMIFLAT MODULES

Abstract

A module P is called F-filtered-projective if for any epimorphism β: B → C and any homomorphism Y: P → C factoring through F, there exists a homomorphism α P → B such that β α = y. We collect for a given module P all such modules F into a class F(P) and all exact sequences relative to which P has the projective property, into a class E(P). Starting with & class P of modules P, we construct the classes F(p) and E(p) as the Intersections of the classes F(P) and E(P) respectively as P runs through P. Relative properties of these classes are investigated and in the special case where P is the class of finitely presented modules, we find a new characterization of flat modules which enables us to introduce the concept of semiflatness which in turn is utilized in a characterization of IF, QF and QF-3 rings.     

Some results on sensitivity analysis in set-valued optimization

In the paper, the higher-order contingent derivative of a parametrized set-valued inclusion is first established. For its applications, we obtain sensitivity analysis of solution map in the decision variable space for a parametrized constrained set-valued optimization problem in terms of higher-order contingent derivatives.