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 Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 1, Pages 71–100 (Mi izv4103)

A saddle-point theorem for strongly and weakly convex functions

G. E. Ivanov

Moscow Institute of Physics and Technology

Abstract: We prove a theorem on the existence, uniqueness, and continuous dependence on parameters for a saddle point in a type of minimax problem that arises, for example, in differential game theory. Our theorem on the existence of a saddle point does not follow from the well-known theorems of von Neumann, Ky Fan, Sion and others since the intersection of sublevel sets of the function considered may be disconnected and non-empty. The hypotheses of our theorem are stated in terms of the strong and weak convexity of functions defined on a Banach space. We study properties of strongly and weakly convex functions related to the operations of minimization and maximization. We obtain unimprovable estimates of convexity parameters for the infimal convolution (episum) and epidifference of functions. This results in the construction of a calculus of convexity parameters of functions with respect to epioperations. We give typical examples and show that the hypotheses of our theorems are essential.

Keywords: saddle point, minimax, strong and weak convexity, differential game.

DOI: https://doi.org/10.4213/im4103

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English version:
Izvestiya: Mathematics, 2011, 75:1, 73–100

Bibliographic databases:

UDC: 517.982.252
MSC: 49J45, 52A41, 26B25, 91A10
Revised: 19.06.2009

Citation: G. E. Ivanov, “A saddle-point theorem for strongly and weakly convex functions”, Izv. RAN. Ser. Mat., 75:1 (2011), 71–100; Izv. Math., 75:1 (2011), 73–100

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv4103
• https://doi.org/10.4213/im4103
• http://mi.mathnet.ru/eng/izv/v75/i1/p71

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This publication is cited in the following articles:
1. Li X.-B., Lin Zh., Wang Q.-L., Chen j.-W., “Holder Continuity of the Saddle Point Set For Real-Valued Functions”, Numer. Funct. Anal. Optim., 38:11 (2017), 1410–1425
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