On Topological Properties of Min-Max Functions

• Published in: Set-valued and variational analysis Vol. 19; no. 2; pp. 237 - 253
• Main Authors: Dorsch, Dominik, Jongen, Hubertus Th, Shikhman, Vladimir
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-06-2011
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Optimization; Closure feasible set; Semi-infinite programming; GSIP; Symmetric Mangasarian-Fromovitz Constraint Qualification; Min-max functions; Lipschitz manifold
• ISSN: 1877-0533; 1877-0541
• DOI: 10.1007/s11228-010-0170-8

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).