We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann’s Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal m...

exists for each x, y ∈ S(E). In this case E is called smooth. The norm of E is said to be Fréchet differentiable if for each x ∈ S(E) the limit is attained uniformly for y ∈ S(E). The normofE is called uniformlyFréchet differentiable if the limit is attained uniformly for x, y ∈ S(E). It is well known that (uniform) Fréchet differentiability of the norm of E implies (uniform) Gâteaux differenti...

In this article we examine to necessary and sufficient optimality conditions for interval optimization problems. We introduce a new concept of stationary point for an interval-valued function based on the gH-derivative. We show the importance the this concept from a practical and computational point of view. We introduce a new concept of invexity for gH-differentiable interval-valued function w...

Nonsmooth operator equations in function spaces are considered, which depend on perturbation parameters. The nonsmoothness arises from a projection onto an admissible interval. Lipschitz stability in L∞ and Bouligand differentiability in L of the parameter-to-solution map are derived. An adjoint problem is introduced for which Lipschitz stability and Bouligand differentiability in L∞ are obtain...

space. For ( ) 1 h L X ∈ ,Γ,μ and Z∈Γ with indicator function ( ) Z I L X ∞ ∈ ,Γ, , μ the integral Z h ∫ dμ will be denoted by Z h I , . Now, we shall define the notion of differentiability for n-set functions. Morris [7] introduced differentiability for set functions and Corley [4] defined this notion for n -set functions. A function : Γ→ φ ! is said to be differentiable at T if there exists (...