Generalized Differentiation and Characterizations for Differentiability of Infimal Convolutions

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• 作者：Nguyen Mau Nam ; Dang Van Cuong
• 关键词：Generalized differentiation ; Distance function ; Minimal time function ; Infimal convolution ; 49J52 ; 49J53 ; 90C31
• 刊名：Set-Valued and Variational Analysis
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：23
• 期：2
• 页码：333-353
• 全文大小：312 KB
• 参考文献：1.Auslender, A.: Differential stability in nonconvex and nondifferentiable programming. Math. Program. 10, 29鈥?1 (1979)MATH
  2.Bauschke, H.H., Wang, X., Ye, J.J., Yuan, X.: Bregman, distance and Chebyshev sets. J. Approx. Theory. 159, 3鈥?5 (2009)View Article MATH MathSciNet
  3.Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis, Springer, CMS Books in Mathematics. Springer, New York (2005)
  4.Bounkhel, M., Thibault, L.: On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48, 223鈥?46 (2002)View Article MATH MathSciNet
  5.Burke, J.V., Ferris, M.C., Qian, M.: On the Clarke subdifferential of the distance function of a closed set. J. Math. Anal. Appl. 166, 199鈥?13 (1992)View Article MATH MathSciNet
  6.Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATH
  7.Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics. Springer, New York (1998)
  8.Colombo, G., Wolenski, P.R.: The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space. J. Global Optim. 28, 269鈥?82 (2004)View Article MATH MathSciNet
  9.Colombo, G., Wolenski, P.R.: Variational analysis for a class of minimal time functions in Hilbert spaces. J. Convex Anal. 11, 335鈥?61 (2004)MATH MathSciNet
  10.De Figueiredo, D.G.: Lectures on the Ekeland variational principle with applications and detours. Springer, West Germany (1989)MATH
  11.Dien, P.H., Yen, N.D.: On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints. Appl. Math. Optim. 24, 35鈥?4 (1991)View Article MATH MathSciNet
  12.He, Y., Ng, K.F.: Subdifferentials of a minimum time function in Banach spaces. J. Math. Anal. Appl. 321, 896鈥?10 (2006)View Article MATH MathSciNet
  13.Jourani, A., Thibault, L., Zagrodny, D.: Differential properties of the Moreau envelope. J. Funct. Anal. 266, 1185鈥?237 (2014)View Article MATH MathSciNet
  14.Jiang, Y., He, Y.: Subdifferentials of a minimum time function in normed spaces. J. Math. Anal. Appl. 358, 410鈥?18 (2009)View Article MATH MathSciNet
  15.Li, C.: On well posedness of best simultaneous approximation problems in Banach spaces. Sci. China Ser A. 12, 1558鈥?570 (2001)View Article
  16.Loewen, P.D.: A mean value theorem for Fr茅chet subgradients. Nonlinear Anal. 23, 1365鈥?381 (1994)View Article MATH MathSciNet
  17.Meng, L., Li, C., Yao, J.-C.: Limiting subdifferentials of perturbed distance functions in Banach spaces. Nonlinear Anal. 75(3), 1483鈥?495 (2012)View Article MATH MathSciNet
  18.Mordukhovich, B.S.: Variational Analysis Generalized, Differentiation. I: Basic Theory, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 330. Springer, Berlin (2006)
  19.Mordukhovich, B.S., Nam, N.M.: Subgradients of minimal time functions under minimal assumptions. J. Convex Anal. 18, 915鈥?47 (2011)MATH MathSciNet
  20.Moreau, J.-J.: Fonctions convexes duales et points proximaux dans un espace Hilbertien. Reports of the Paris Academy of Sciences, Series A 255, 2897鈥?899 (1962)MATH MathSciNet
  21.Nam, N.M., Villalobos, M.C., An, N.T.: Minimal time functions and the smallest Intersecting ball problem with unbounded dynamics. J. Optim. Theory Appl. 154, 768鈥?91 (2012)View Article MATH MathSciNet
  22.Nam, N.M.: Subdifferential formulas for a class of nonconvex infimal convolution. e-print (2014)
  23.Nesterov, Yu.: Introductory Lectures on Convex Optimization. A Basic Course (2004)
  24.Wang, B.: Beginning Variational Analysis (lecture note), Bohai University, Jinzhou (2010)
  25.Wu, Z., Ye, J.J.: Equivalences among various derivatives and subdifferentials of the distance function. J. Math. Anal. Appl. 282, 629鈥?47 (2003)View Article MATH MathSciNet
  26.Zhang, Y., He, Y., Jiang, Y.: Subdifferentials of a perturbed minimal time function in normed spaces. Optim. Lett. 8, 1921鈥?930 (2014)
  
• 作者单位：Nguyen Mau Nam (1)
  Dang Van Cuong (2)
  
  1. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, PO Box 751, Portland, OR, 97207, USA
  2. Department of Mathematics, Faculty of Natural Sciences, Duy Tan University, K7/25 Quang Trung, Da Nang, Vietnam
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1877-0541

文摘

This paper is devoted to the study of generalized differentiation properties of the infimal convolution. This class of functions covers a large spectrum of nonsmooth functions well known in the literature. The subdifferential formulas obtained unify several known results and allow us to characterize the differentiability of the infimal convolution which plays an important role in variational analysis and optimization.