Generalized Differential Calculus for Nonsmooth and Set-Valued Mappings

A Sard Theorem for Set-Valued Mappings∗†

If F is a set-valued mapping from IRn into IRm with closed graph, then y ∈ IRm is a critical value of F if for some x with y ∈ F (x), F is not metrically regular at (x, y). We prove that the set of critical values of a set-valued mapping whose graph is a definable (tame) set in an o-minimal structure containing additions and multiplications is a set of dimension not greater than m − 1 (resp. a ...

Fixed Point Theorems for Set-valued Mappings

On generalized Ekeland's variational principle and equivalent formulations for set-valued mappings

We propose a very weak type of generalized distance called weak τ -function and use it to weaken the assumptions about lower semicontinuity in existing formulations of Ekeland’s variational principle for a kind of minimizers of a set-valued mapping, which is different from the Pareto minimizer, and in recent results which are equivalent to Ekeland’s variational principle.

Optimality conditions for Pareto efficiency and proper ideal point in set-valued nonsmooth vector optimization using contingent cone

In this paper, we first present a new important property for Bouligand tangent cone (contingent cone) of a star-shaped set. We then establish optimality conditions for Pareto minima and proper ideal efficiencies in nonsmooth vector optimization problems by means of Bouligand tangent cone of image set, where the objective is generalized cone convex set-valued map, in general real normed spaces.

Subdifferentials of Performance Functions and Calculus of Coderivatives of Set-valued Mappings

The paper contains calculus rules for coderivatives of compositions, sums and intersections of set-valued mappings. The types of coderivatives considered correspond to Dini-Hadamard and limiting Dini-Hadamard subdifferentials in Gâteaux differentiable spaces, Fréchet and limiting Fréchet subdifferentials in Asplund spaces and approximate subdifferentials in arbitrary Banach spaces. The key elem...