文摘
Chen and Mangasarian (Comput Optim Appl 5:97-38, 1996) developed smoothing approximations to the plus function built on integral-convolution with density functions. X. Chen (Math Program 134:71-9, 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 (Math Program 134:71-9, 2012). In particular, this guarantees the desired limiting behavior of critical points of the smooth approximations.