Optimization of Set-Valued Functions

K-epiderivatives for set-valued functions and optimization

Monotone set-valued functions defined by set-valued Choquet integrals

In this paper, some properties of the monotone set-valued function defined by the set-valued Choquet integral are discussed. It is shown that several important structural characteristics of the original set function, such as null-additivity, strong order continuity, property(S) and pseudometric generating property, etc., are preserved by the new set-valued function. It is also shown that integr...

Slope and G-set Characterization of Set-valued Functions and Applications to Non-differentiable Optimization

In this paper we derive a generalizing concept of G-norms, which we call G-sets, which is used to characterize minimizers of non-differentiable regularization functionals. Moreover, the concept is closely related to the definition of slopes as published in a recent book by Ambrosio, Gigli, Savaré. A paradigm of regularization models fitting in this framework is robust bounded variation regulari...

Approximation of Set-valued Functions by Single-valued One

Let Σ : M → 2 \ {∅} be a set-valued function defined on a Hausdorff compact topological space M and taking values in the normed space (Y, ‖·‖). We deal with the problem of finding the best Chebyshev type approximation of the set-valued function Σ by a singlevalued function g from a given closed convex set V ⊂ C(M, Y ). In an abstract setting this problem is posed as the extremal problem supt∈M ...

Pseudo-integral of Set-valued Functions

An approach to the integration of set-valued functions based on the pseudo-integral has been proposed. Some basic properties of the pseudo-integral of set-valued functions have been shown.