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
Ke Yin
Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China
kyin@hust.edu.cn
Kewei Zhang
School of Mathematical Sciences, University of Nottingham, United Kingdom
kewei.zhang@nottingham.ac.uk
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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