On Topological Properties of Min-Max Functions

We examine the topological structure of the upper-level set M ^max given by a min-max function φ. It is motivated by recent progress in Generalized Semi-Infinite Programming (GSIP). Generically, M ^max is proven to be the topological closure of the GSIP feasible set (see Guerra-Vázquez et al. 2009; Günzel et al., Cent Eur J Oper Res 15(3):271–280, 2007). We formulate two assumptions (Compactness Condition CC and Sym-MFCQ) which imply that M ^max is a Lipschitz manifold (with boundary). The Compactness Condition is shown to be stable under C ⁰-perturbations of the defining functions of φ. Sym-MFCQ can be seen as a constraint qualification in terms of Clarke’s subdifferential of the min-max function φ. Moreover, Sym-MFCQ is proven to be generic and stable under C ¹-perturbations of the defining functions which fulfill the Compactness Condition. Finally we apply our results to GSIP and conclude that generically the closure of the GSIP feasible set is a Lipschitz manifold (with boundary).