On Topological Properties of Min-Max Functions |
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Authors: | Dominik Dorsch Hubertus Th Jongen Vladimir Shikhman |
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Institution: | (1) Inst. OR Univ. Zurich, Zurich, Switzerland |
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Abstract: | 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). |
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