Journal of Convex Analysis 25 (2018), No. 2, 389--402
Copyright Heldermann Verlag 2018
On Minimax Theorems for Lower Semicontinuous Functions in Hilbert Spaces
Ewa Bednarczuk
System Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland
and: Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland
Ewa.Bednarczuk@ibspan.waw.pl
Monika Syga
Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland
M.Syga@mini.pw.edu.pl
We prove minimax theorems for lower semicontinuous functions defined on a Hilbert space. The main tools are the theory of Φ-convex functions and sufficient and necessary conditions for the minimax equality for general Φ-convex functions. The conditions we propose are expressed in terms of abstract Φ-subgradients.
Keywords: Abstract convexity, minimax theorems, intersection property, abstract Phi-subdifferential, abstract Phi-subgradient.
MSC: 32F17, 49J52, 49K27, 49K35, 52A01
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