Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.

Stationarity and regularity of infinite collections of sets. Applications to infinitely constrained optimization

This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López (J. Optim. Theory Appl. 154(2), 2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optimization problems. We consider several settings of optimization problems which involve (explicitly or implicitly) infinite collections of sets and deduce for them necessary conditions characterizing stationarity in terms of dual space elements—normals and/or subdifferentials.