Gradient Consistency for Integral-convolution Smoothing Functions

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• 作者：James V. Burke (1)
  Tim Hoheisel (2)
  Christian Kanzow (2)
  
• 关键词：Smoothing method ; Subdifferential calculus ; Integral ; convolution ; Piecewise ; affine function ; 49J52 ; 90C26 ; 90C30
• 刊名：Set-Valued and Variational Analysis
• 出版年：2013
• 出版时间：June 2013
• 年：2013
• 卷：21
• 期：2
• 页码：359-376
• 全文大小：403KB
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• 作者单位：James V. Burke (1)
  Tim Hoheisel (2)
  Christian Kanzow (2)
  
  1. Department of Mathematics, University of Washington, P.O. Box 354350, Seattle, WA, 98195-4350, USA
  2. Institute of Mathematics, University of Würzburg, Campus Hubland Nord, Emil-Fischer-Stra?e 30, 97074, Würzburg, Germany
  
• ISSN：1877-0541

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.