Chen and Mangasarian (1995) developed smoothing approximations to the plus function built on integral-convolution with density functions. X. Chen (2012) has recently picked up this idea constructing a large class of smoothing functions for nonsmooth minimization through composition with smooth mappings. In this paper, we generalize this idea by substituting the plus function for an arbitrary finite max-function. Calculus rules such as inner and outer composition with smooth mappings are provided, showing that the new class of smoothing functions satisfies, under reasonable assumptions, gradient consistency, a fundamental concept coined by Chen (2012). In particular, this guarantees the desired limiting behavior of critical points of the smooth approximations.
Preprint 309, Institute of Mathematics, University of Würzburg, Würzburg, September 2012.