About contingent epiderivatives

In this paper we analyze some questions concerning the contingent epiderivative. When the ordering cone is not necessarily pointed we introduce the notion of family of contingent epiderivatives. We also investigate the relationship between the contingent derivative and the generalized epiderivative. Furthermore, we give existence theorems with respect to Daniell cones. As a particular case, we study set-valued maps that take values in Rn.     

Variational characterization of the contingent epiderivative

In this paper the existence of the contingent epiderivative of a set-valued map is studied from a variational perspective. We give a variational characterization of the ideal minimal of a weakly compact set. As a consequence we characterize the existence of the contingent epiderivative in terms of an associated family of variational systems. When a set-valued map takes values in Rn we show that these systems can be formulated in terms of the contingent epiderivatives of scalar set-valued maps. By applying these results we extend some existing theorems.     

A chain rule for first- and second-order epiderivatives and hypoderivatives

A chain rule is established for contingent and adjacent epiderivatives and hypoderivatives of compositions g○h, where h is assumed to be Hadamard directionally differentiable. Corollaries include a formula for the contingent and adjacent cones of an equality constraint set defined by a Hadamard directionally differentiable function. An analogous chain rule for second-order contingent and adjacent epiderivatives and hypoderivatives is also developed.     

Scalarization and optimality conditions for strict minimizers in multiobjective optimization via contingent epiderivatives

In this paper necessary and sufficient conditions for strict minimizers of a general vector optimization problem are given by means of different notions of graphical or epigraphical derivatives, extending some existing results in the literature. In first place, these conditions are established by means of contingent derivatives and secondly, by applying a nonconvex separation result, in terms of contingent epiderivatives and hypoderivatives. Moreover, through a variational approach, a scalarization method is developed in order to obtain scalar versions of these results.     

Some Calculus Rules for Contingent Epiderivatives

This article presents some calculus rules for contingent epiderivatives of set-valued maps. Among other results the main emphasis is focused on a formula for scalar multiplication, sum formulae and chain rules. The calculus of contingent cones and some inversion theorems are used as a tool. Some applications are also given.     

Epidifferentiability of the map of infima in Hilbert spaces

This article deals with derivatives for set-valued maps that take values in ordered vector spaces, in particular it concerns about the relationship between the epiderivatives of a set-valued map and its associated map of infima. When the image space is a real separable Hilbert space ordered by an orthonormal basis, by using a variational technique based on a decoupling of the ordering cone into half-spaces, we show that both epiderivatives coincide under certain hypothesis of compactness and stability. Furthermore we obtain some computation formulas for these derivatives in terms of associated scalar set-valued maps.     

Generalized second-order contingent epiderivatives in parametric vector optimization problems

This paper is concerned with generalized second-order contingent epiderivatives of frontier and solution maps in parametric vector optimization problems. Under some mild conditions, we obtain some formulas for computing generalized second-order contingent epiderivatives of frontier and solution maps, respectively. We also give some examples to illustrate the corresponding results.     

-contingent epiderivatives in reflexive spaces

In this paper we introduce a notion of the τ^w-contingent epiderivative of a set-valued map by considering the weak topology in the image space. We establish a chain rule formula for families of τ^w-contingent and contingent epiderivatives. When the image space is reflexive and the ordering cone a half-space, we give conditions for the existence of the family of τ^w-contingent epiderivatives of a stable set-valued map; moreover we obtain an interesting formula that connects this family with the contingent epiderivative of an associated scalar set-valued map. Finally we apply the previous results in order to establish optimality conditions for a set-valued optimization problem via epiderivatives with respect to half-spaces associated with the ordering cone. In particular we provide a scalarization method for computing these conditions in reflexive spaces.     

Optimality Conditions for Vector Equilibrium Problems in Terms of Contingent Epiderivatives

In this article, we study some important properties of contingent epiderivatives concerning steady functions and a cone with a compact base along with its applications to establish necessary and sufficient optimality conditions for weakly efficient, Henig efficient, globally efficient and superefficient solutions for no constraints and constraints (it concludes cone constraint, equality constraint and a constraint set) vector equilibrium problems in terms of contingent epiderivatives. We also give some examples to illustrate obtained results.     

The role of directional compactness in the existence and computation of contingent epiderivatives

This paper underlines the role of directional compactness in the scalarization of graphical derivatives of set-valued maps taking values in infinite-dimensional spaces. Two main theorems are given. The first one states the equivalence of contingent epiderivatives and τw-contingent epiderivatives for directionally compact maps. The second main result proves a variational characterization for the contingent epiderivative of stable and directionally compact maps taking values in general image spaces, extending known results in finite-dimensional and reflexive Banach spaces. The hypotheses given are minimal as is shown by means of several examples. Connections of these theorems with other results of the literature are also provided.     

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.     

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.     

Calmness Properties and Contingent Subgradients of Integral Functionals on Lebesgue Spaces L p , 1 ≼ p < ∞

We study the notions of calmness, contingent epiderivatives, Hadamard differentiability, Gateaux differentiability, and contingent subdifferentials of an integral functional defined on a Lebesgue space L _p , with p ≠ ∞.     

Second-Order Optimality Conditions for Set-Valued Vector Equilibrium Problems

In this article, by using the generalized second-order contingent (adjacent) epiderivatives of set-valued maps, we obtain necessary optimality conditions and sufficient optimality conditions for weakly efficient solutions, Henig efficient solutions to the set-valued vector equilibrium problems with constraints. Some results of this article improve the corresponding results in literatures by lessening the assumption of convexity.     

不完备证券市场中博弈未定权益的保值

考虑不完备证券市场中博弈未定权益(GCC)的保值问题,通过Kramkov关于上鞅的可选分解定理给出未定权益的上保值价格和下保值价格。指出关于买卖双方都存在着一个最优保值策略。给出价格的一个无套利区间,并针对前面的结论,给出几个性质以及在限制投资组合方面的一个应用。     

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 the uniform boundedness of solutions of contingent equations

It is proved that, under standard conditions (in particular the convexity of the velocity set), the boundedness of the individual solutions of a contingent equation implies the uniform boundedness of the solutions, and consequently the boundedness of the reachable sets. This result is used to obtain an existence theorem in optimal control theory. By means of an example, it is shown that the convexity condition cannot be omitted.     

Martingale Representation and Admissible Portfolio Process with Regime Switching

We discuss the existence of an admissible investment strategy for any given consumption rate process in a Markov, regime-switching Black–Scholes–Merton economy. A martingale representation for a double martingale generated by the Brownian motion and the Markov chain is used to establish the existence of the admissible investment strategy. We also employ the martingale representation to prove the attainability of a European contingent claim in the regime-switching environment under a pricing kernel specified by the Esscher transform based on the Laplace cumulant process.     

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.     

随机限制市场中的最优溢价

本文讨论不定权益在一簇辅助市场中的上确界的到达性 ,该问题与限制市场中不定权益的对冲问题密切相关