Gradient Consistency for Integral-convolution Smoothing Functions

James V. Burke; Tim Hoheisel; Christian Kanzow

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Gradient Consistency for Integral-convolution Smoothing Functions

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Chen and Mangasarian (Comput Optim Appl 5:97-138, 1996) developed smoothing approximations to the plus function built on integral-convolution with density functions. X. Chen (Math Program 134:71-99, 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 134:71-99,2012). In particular, this guarantees the desired limiting behavior of critical points of the smooth approximations.

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《Artificial intelligence and law》 |2013年第2期|359-376|共18页
作者
James V. Burke; Tim Hoheisel; Christian Kanzow;
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作者单位

Department of Mathematics, University of Washington, P.O. Box 354350, Seattle, WA 98195-4350, USA;

Institute of Mathematics, University of Wuerzburg, Campus Hubland Nord, Emil-Fischer-Strasse 30, 97074 Wuerzburg, Germany;

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正文语种英语
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Smoothing method; Subdifferential calculus; Integral-convolution; Piecewise-affine function;

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Gradient Consistency for Integral-convolution Smoothing Functions

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