Applying twice a minimax theorem

2019/07/16 by Ricceri, Biagio
#FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1907.07016

Abstract

Here is one of the results obtained in this paper: Let X, Y be two convex sets each in a real vector space, let J:X× Y→ \bf R be convex and without global minima in X and concave in Y, and let Φ:X→ \bf R be strictly convex. Also, assume that, for some topology on X, Φ is lower semicontinuous and, for each y∈ Y and λ>0, J(⋅,y) is lower semicontinuous and J(⋅,y)+λΦ(⋅) is inf-compact. Then, for each r∈ ]inf_XΦ,sup_XΦ[ and for each closed set S⊆ X satisfying Φ^-1(r)⊆ S⊆ Φ^-1(]-∞,r]) , one has sup_Yinf_SJ=inf_Ssup_YJ .

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