On Frèchet normal cone for nonsmooth mathematical programming problems with switching constraints

Abstract

This paper is devoted to the study of a class of nonsmooth programming problems with switching constraints (abbreviated as, (NMPSC)), where all the involved functions in the switching constraints are assumed to be locally Lipschitz. We investigate the properties of Frèchet normal cone of (NMPSC). In particular, we introduce two Guignard type constraint qualifications for (NMPSC) in terms of Michel-Penot subdifferential. Moreover, we derive two estimates for the Frèchet normal cone of (NMPSC) and further establish stationarity conditions at an optimal solution for (NMPSC). To the best of our knowledge, this is for the first time Frèchet normal cone for (NMPSC) have been studied in the setting of Euclidean spaces.