Journal of Convex Analysis 29 (2022), No. 4, 1193--1224
Copyright Heldermann Verlag 2022
A Tight Smooth Approximation of the Maximum Function and its Applications
Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China
School of Mathematical Sciences, University of Nottingham, United Kingdom
We analyse the C1,1 tight approximations of the finite maximum function defined by the upper compensated convex transform introduced in a previous paper of the second author [ Compensated convexity and its applications, Ann. Inst. H. Poincaré (C), Non Linear Analysis 25/4 (2008) 743--771]. We present the precise geometric structure, the tightness property, the sharp error estimates and the asymptotic properties of our approximation. We compare our method with the well-known "log-sum-exp" smooth approximation by showing that our approximation is geometrically much sharper than the "log-sum-exp" approximation. We apply our results to smooth approximations for functions defined by the maximum of finitely many smooth functions in Rn arising from finite and semi-infinite minimax optimization problems.
Keywords: Maximum function, compensated convex transforms, tight smooth approximation, convex optimization, minimax problem.
MSC: 90C25, 90C30, 90C34, 90C47, 65K05, 65K10, 49M20, 49M37.
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