
Journal of Convex Analysis 29 (2022), No. 4, 11931224 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 C^{1,1} 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) 743771]. 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 wellknown "logsumexp" smooth approximation by showing that our approximation is geometrically much sharper than the "logsumexp" approximation. We apply our results to smooth approximations for functions defined by the maximum of finitely many smooth functions in R^{n} arising from finite and semiinfinite 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. [ Fulltextpdf (232 KB)] for subscribers only. 