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A minimax theorem. (English) Zbl 0572.49003
The paper centers around minimax problems of the form \(\min_{x\in A}\max_{y\in B}H(x,y)\) for \(H: X\times Y\to [-\infty,\infty]\) where A and B are nonempty sets in topological vector spaces X and Y. The author investigates the case in which H is a saddle or a quasi-saddle function and A, B are convex not necessarily compact sets. Derivation of the presented minimax theorems relies on a lemma established by Ky Fan [Math. Ann. 142, 305-310 (1961; Zbl 0093.367)]. The results are applied to the Fenchel-Rockafellar duality model for constrained convex minimization problems. Besides, saddlepoint criteria are collected in the general setting of closed saddle functions and implications among them are explored.
Reviewer: M.Vlach

MSC:
49J35 Existence of solutions for minimax problems
49N15 Duality theory (optimization)
90C25 Convex programming
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
49J45 Methods involving semicontinuity and convergence; relaxation
90C48 Programming in abstract spaces
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