This article is cited in 1 scientific paper (total in 1 paper)
A saddle-point theorem for strongly and weakly convex functions
G. E. Ivanov
Moscow Institute of Physics and Technology
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.
saddle point, minimax, strong and weak convexity, differential game.
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Izvestiya: Mathematics, 2011, 75:1, 73–100
MSC: 49J45, 52A41, 26B25, 91A10
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
\paper A saddle-point theorem for strongly and weakly convex functions
\jour Izv. RAN. Ser. Mat.
\jour Izv. Math.
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This publication is cited in the following articles:
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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