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Gradient consistency for integral-convolution smoothing functions

James V. Burke(burke***at***math.washington.edu)
Tim Hoheisel(hoheisel***at***mathematik.uni-wuerzburg.de)
Christian Kanzow(kanzow***at***mathematik.uni-wuerzburg.de)

Abstract: 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.

Keywords: smoothing method, subdifferential calculus, integral-convolution, piecewise-affine function

Category 1: Convex and Nonsmooth Optimization

Category 2: Nonlinear Optimization

Citation: Preprint 309, Institute of Mathematics, University of Würzburg, Würzburg, September 2012.

Download: [PDF]

Entry Submitted: 09/08/2012
Entry Accepted: 09/08/2012
Entry Last Modified: 09/08/2012

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