A Tight Smooth Approximation of the Maximum Function and its Applications

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) 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 $\mathbb{R}^n$ arising from finite and semi-infinite minimax optimization problems.