A Tight Smooth Approximation of the Maximum Function and its Applications

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. Poincare (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.