Abstract
In this paper, we consider a class of mathematical programs with switching constraints (MPSCs) where the objective involves a non-Lipschitz term. Due to the non-Lipschitz continuity of the objective function, the existing constraint qualifications for local Lipschitz MPSCs are invalid to ensure that necessary conditions hold at the local minimizer. Therefore, we propose some MPSC-tailored qualifications which are related to the constraints and the non-Lipschitz term to ensure that local minimizers satisfy the necessary optimality conditions. Moreover, we study the weak, Mordukhovich, Bouligand, strongly (W-, M-, B-, S-) stationay, analyze what qualifications making local minimizers satisfy the (M-, B-, S-) stationay, and discuss the relationship between the given MPSC-tailored qualifications. Finally, an approximation method for solving the non-Lipschitz MPSCs is given, and we show that the accumulation point of the sequence generated by the approximation method satisfies S-stationary under the second-order necessary condition and MPSC Mangasarian-Fromovitz (MF) qualification.
Highlights
We show that any accumulation point of the sequence generated by our method is W-stationary under mathematical programs with switching constraints (MPSCs)-MF qualification, and is S-stationary under second-order necessary condition and MPSC Mangasarian-Fromovitz (MPSC-MF) qualification
We show that xis an S-stationary point of problem (1) when MPSC linear independence (MPSC-LI)
In the proof of Theorem 7, we have shown that xis a W-stationary point under the MPSC-MF qualification condition
Summary
In order to facilitate the reading of this article, some notations are given first: N. Given a nonempty closed set A and a point x ∗ ∈ A. (Limiting normal cone) The limiting normal cone of A at x ∗ is defined as b A ( x k ), s.t. dk → d}. Given a continuous function ψ : Rs → R and a point x ∈ Rs , we recall the definitions of regular subdifferential, limiting subdifferential, and horizon subdifferential [28]. (Regular subdifferential) The regular subdifferential of ψ at xis defined as b. (Limiting subdifferential) The limiting subdifferential of ψ at xis defined as ( x k ), s.t. υk → υ}.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call