Subdifferentiation of Regularized Functions

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• 作者：Huynh Van Ngai ; Jean-Paul Penot
• 关键词：Approximately convex function ; Convolution ; Monotonicity ; Nonsmooth analysis ; Regularization ; Subdifferential ; 49J52 ; 46N10 ; 46T20
• 刊名：Set-Valued and Variational Analysis
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：24
• 期：1
• 页码：167-189
• 全文大小：402 KB
• 参考文献：1.Asplund, E.: Fréchet differentiability of convex functions. Acta Math. 121, 31–47 (1968)CrossRef MathSciNet MATH
  2.Attouch, H.: Variational Convergence of Functions and Operators. Pitman, London (1984)
  3.Attouch, H., Azé, D.: Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method. Ann. Inst. Henri Poincaré 10(3), 289–312 (1993)MATH
  4.Aubin, J.-P., Frankowska, H.: Set-valued Analysis. Birkhaüser, Boston (1990)MATH
  5.Aussel, D., Daniilidis, A., Thibault, L.: Subsmooth sets: functional characterizations and related concepts. Trans. Am. Math. Soc. 357(4), 1275–1301 (2005)CrossRef MathSciNet MATH
  6.Benoist, J.: Convergence de la dérivée de la régularisée de Lasry-Lions. C. R. Acad. Sci. Paris 315, 941–944 (1992)MathSciNet MATH
  7.Benyamini, Y., Lindenstrauss, J.: Geometric nonlinear functional analysis, amer. Math. Soc. Colloquium publications 48 providence (2000)
  8.Bernard, F., Thibault, L.: Prox-regularity of functions and sets in Banach spaces. Set-Valued Anal. 12(1-2), 25–47 (2004)CrossRef MathSciNet MATH
  9.Bernard, F., Thibault, L.: Prox-regular functions in Hilbert spaces. J. Math. Anal. Appl. 303(1), 1–14 (2005)CrossRef MathSciNet MATH
  10.Bernard, F., Thibault, L.: Uniform prox-regularity of functions and epigraphs in Hilbert spaces. Nonlinear Anal. 60(2), 187–207 (2005)CrossRef MathSciNet MATH
  11.Bernard, F., Thibault, L., Zlateva, N.: Characterizations of prox-regular sets in uniformly convex Banach spaces. J. Convex Anal. 13(3-4), 525–559 (2006)MathSciNet MATH
  12.Bernard, F., Thibault, L., Zlateva, N.: Prox-regular sets and epigraphs in uniformly convex Banach spaces: various regularities and other properties. Trans. Amer. Math. Soc. 363(4), 2211–2247 (2011)CrossRef MathSciNet MATH
  13.Bernard, F., Thibault, L., Zagrodny, D.: Integration of primal lower nice functions in Hilbert spaces. J. Optim. Theory Appl. 124(3), 561–579 (2005)CrossRef MathSciNet MATH
  14.Borwein, J.M., Fitzpatrick, S.: Existence of nearest points in Banach spaces. Can. J. Math.(XLI) 4, 702–720 (1989)CrossRef MathSciNet
  15.Borwein, J.M., Giles, J.: The proximal normal formula in Banach space. Trans. Amer. Math. Soc. 302(1), 371–381 (1987)CrossRef MathSciNet MATH
  16.Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis Canadian Mathematical Society Books in Maths, vol. 20. Springer, New York (2005)
  17.Bougeard, M., Penot, J.-P., Pommellet, A.: Towards minimal assumptions for the infimal convolution regularization. J. Approximation Theory 64(3), 245–272 (1991)CrossRef MathSciNet MATH
  18.Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)MATH
  19.Clarke, F.H.: Functional Analysis, Calculus of Variations and Optimal Control Graduate Texts in Mathematics, vol. 264. Springer, London (2013)CrossRef
  20.Clarke, F.H., Stern, R.J., Wolenski, P.R.: Proximal smoothness and the lower-C 2 property. J. Convex Anal. 2(1,2), 117–144 (1995)MathSciNet MATH
  21.Colombo, G., Goncharov, V.: Variational inequalities regularity properties of closed sets in Hilbert spaces. J. Convex Anal. 8, 197–221 (2001)MathSciNet MATH
  22.Colombo, G., Thibault, L.: Prox-regular sets and applications. Handbook of nonconvex analysis and applications, pp. 99–182. International press, Somerville (2010)
  23.Correa, R., Jofre, A., Thibault, L.: Subdifferential monotonicity as a characterization of convex functions. Numer. Funct. Anal. Optim. 15, 531–535 (1994)CrossRef MathSciNet MATH
  24.Combari, C., Poliquin, R.A., Thibault, L.: Convergence of subdifferentials of convexly composite functions. Canad. J. Math. 51, 250–265 (1999)CrossRef MathSciNet MATH
  25.Daniilidis, A., Georgiev, P.: Approximate convexity and submonotonicity. J. Math. Anal. Appl. 291, 292–301 (2004)CrossRef MathSciNet MATH
  26.Daniilidis, A., Georgiev, P., Penot, J.-P.: Integration of multivalued operators and cyclic submonotonicity. Trans. Amer. Math. Soc. 355, 177–195 (2003)CrossRef MathSciNet MATH
  27.Degiovanni, M., Marino, A., Tosques, M.: Evolution equations with lack of convexity. Nonlinear Anal. 9, 1401–1443 (1985)CrossRef MathSciNet MATH
  28.Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)CrossRef MathSciNet MATH
  29.Diestel, J.: Geometry of banach spaces, select topics, lecture notes in math, vol. 485. Springer (1975)
  30.Fabian, M.: Subdifferentiability and trustwothiness in the light of a new variational principle of Borwein and Preiss. Acta Univ. Carolinae 30, 51–56 (1989)
  31.Georgiev, P., Zlateva, N.P.: Reconstruction of the Clarke subdifferential by Lasry-Lions regularizations. J. Math. Anal. Appl. 248, 415–428 (2000)CrossRef MathSciNet MATH
  32.Holmes, R.B.: Geometric Functional Analysis and Its Applications Graduate Texts in Maths, vol. 24. Springer, New York (1975)CrossRef
  33.Ioffe, A.D.: Subdifferentiability spaces and nonsmooth analysis. Bull. Am. Math. Soc., New Ser. 10, 87–90 (1984)CrossRef MathSciNet MATH
  34.Ioffe, A.D.: On subdifferentiability spaces. New York Acad. Sci. 410, 107–119 (1983)CrossRef MathSciNet
  35.Ioffe, A.D.: Proximal analysis and approximate subdifferentials. J. London Math. Soc. 41, 175–192 (1990)CrossRef MathSciNet MATH
  36.Ivanov, M., Zlateva, N.: On the primal lower nice property. C. R. Acad.. Bulgare Sci. 54(11), 5–10 (2001)MathSciNet MATH
  37.Jourani, A.: Limit superior of subdifferentials of uniformly convergent functions. Positivity 3, 33–47 (1999)CrossRef MathSciNet MATH
  38.Jourani, A., Thibault, L.: Metric inequality and subdifferential calculus in Banach spaces. Set-valued Anal. 3, 87–100 (1995)CrossRef MathSciNet MATH
  39.Jourani, A., Thibault, L., Zagrodny, D.: C 1,ω-regularity and Lipschitz-like properties of subdifferential. Proc. Lond. Math. Soc. (3) 1, 189–223 (2012)CrossRef MathSciNet
  40.Jourani, A., Thibault, L., Zagrodny, D.: Differential properties of the Moreau envelope. J. Funct. Anal. 266(3), 1185–1237 (2014)CrossRef MathSciNet MATH
  41.Kruger, A.Y., Mordukhovich, B.S.: Extremal points the Euler equation in nonsmooth optimization problems. Dokl. Akad Nauk BSSR 24, 684–687 (1980)MathSciNet MATH
  42.Lasry, J.-M., Lions, P.-L.: A remark on regularization in Hilbert spaces. Israel J. Math. 55, 257–266 (1986)CrossRef MathSciNet MATH
  43.Levy, A.B., Poliquin, R.A., Thibault, L.: Partial extensions of Attouch’s theorem with applications to proto-derivatives of subgradient mappings. Trans. Amer. Math. Soc. 347(4), 1269–1293 (1995)MathSciNet MATH
  44.Lopez, O., Thibault, L.: Sequential formula for subdifferential of upper envelope of convex functions. J. Nonlinear Convex Anal. 14(2), 377–388 (2013)MathSciNet MATH
  45.Luc, D.T., Ngai, H.V., Théra, M.: On ε−convexity and ε-monotonicity, in Calculus of Variations and Differential Equations. In: Ioffe, A., Reich, S., Shafrir, I. (eds.) Research Notes in Maths, pp 82–100. Chapman & Hall (1999)
  46.Marcellin, S., Thibault, L.: Evolution problems associated with primal lower nice functions. J. Convex Anal. 13(2), 385–421 (2006)MathSciNet MATH
  47.Marino, A., Tosques, M.: Some variational problems with lack of convexity and some partial differential inequalities, Methods of Nonconvex Analysis, Lect. 1st Sess. CIME, Varenna/Italy 1989. Lect. Notes Math. 1446, 58–83 (1990)CrossRef MathSciNet
  48.Mazade, M., Thibault, L.: Differential variational inequalities with locally prox-regular sets. J. Convex Anal. 19(4), 1109–1139 (2012)MathSciNet MATH
  49.Mazade, M., Thibault, L.: Regularization of differential variational inequalities with locally prox-regular sets. Math. Program. 139(1–2), 243–269 (2013). Ser. BCrossRef MathSciNet MATH
  50.Mazade, M., Thibault, L.: Primal Lower Nice Functions and Their Moreau Envelopes. Computational and Analytical Mathematics, 521–553, Springer Proc. Math Stat., vol. 50. Springer, New York (2013)
  51.Moreau, J.-J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)MathSciNet MATH
  52.Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348(4), 1235–1280 (1996)CrossRef MathSciNet MATH
  53.Mordukhovich, B.: Variational analysis and generalized differentiation. I. Basic Theory Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer, Berlin (2006)
  54.Mordukhovich, B.: Variational analysis and generalized differentiation. II. Applications. Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 331. Springer, Berlin (2006)
  55.Ngai, H.V., Luc, D.T., Théra, M.: Approximate convex functions. J. Nonlinear and Convex Anal. 1(2), 155–176 (2000)MathSciNet MATH
  56.Ngai, H.V., Penot, J.-P.: Approximately convex functions and approximately monotone operators. Nonlinear Anal. 66, 547–564 (2007)CrossRef MathSciNet MATH
  57.Ngai, H.V., Penot, J.-P.: Paraconvex functions and paraconvex sets. Stud. Math. 184(1), 1–29 (2008)CrossRef MathSciNet MATH
  58.Ngai, H.V., Penot, J.-P.: Approximately convex sets. J. Nonlinear and Convex Anal. 8(3), 337–371 (2007)MathSciNet MATH
  59.Ngai, H., Théra, M.: Metric inequality, subdifferential calculus and applications. Set-Valued Anal. 9, 187–216 (2001)CrossRef MathSciNet MATH
  60.Penot, J.-P.: Proximal Mappings. J. Approx. Theory 94, 203–221 (1998)CrossRef MathSciNet MATH
  61.Penot, J.-P.: Favorable classes of mappings and multimappings in nonlinear analysis and optimization. J. Convex Analysis 3, 97–116 (1996)MathSciNet MATH
  62.Penot, J.-P.: Calmness and stability properties of marginal and performance functions. Numer. Functional Anal. Optim. 25(3–4), 287–308 (2004)MathSciNet MATH
  63.Penot, J.-P.: Differentiability properties of optimal value functions. Canadian J. Math. 56(4), 825–842 (2004)CrossRef MathSciNet MATH
  64.Penot, J.-P.: Softness, sleekness and regularity properties in nonsmooth analysis. Nonlinear Anal. 68(9), 2750–2768 (2008)CrossRef MathSciNet MATH
  65.Penot, J.-P.: Calculus without Derivatives Graduate Texts in Mathematics, vol. 266. Springer, New York (2013)CrossRef
  66.Penot, J.-P.: Analysis. From Concepts to Applications, Universitext. Springer, London. (to appear)
  67.Penot, J.-P., Ratsimahalo, R.: On the Yosida approximation of operators. Proc. Royal Soc. Edinburg 131A, 945–966 (2001)CrossRef MathSciNet
  68.Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability, Lect. Notes in Math., vol. 1364. Springer, Berlin (1993). (second edition)
  69.Poliquin, R.A.: Integration of subdifferentials of nonconvex functions. Nonlinear Anal. 17, 385–398 (1991)CrossRef MathSciNet MATH
  70.Poliquin, R.A.: An extension of Attouch’s theorem and its application to second-order epi-differentiation of convexly composite functions. Trans. Amer. Math. Soc. 322, 861–874 (1991)MathSciNet
  71.Poloquin, R.A., Rockafellar, R.T.: Prox-regular functions in variational analysis 348, 1805–1818 (1995)
  72.Poloquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 307, 5231–5249 (2000)CrossRef
  73.Rockafellar, R., Wets, R.J.-B.: Variational Analysis, Grundlehren Der Mathematishen Wissenschaften, vol. 317. Springer, Berlin (2002)
  74.Rolewicz, S.: On the coincidence of some subdifferentials in the class of α(⋅)-paraconvex functions. Optimization 50, 353–360 (2001)CrossRef MathSciNet MATH
  75.Spingarn, J.E.: Submonotone subdifferentials of Lipschitz functions. Trans. Amer. Math. Soc. 264, 77–89 (1981)CrossRef MathSciNet MATH
  76.Sebbah, M., Thibault, L.: Metric projection and compatibly parameterized families of prox-regular sets in Hilbert space. Nonlinear Anal. 75(3), 1547–1562 (2012)CrossRef MathSciNet MATH
  77.Serea, O., Thibault, L.: Primal-lower-nice property of value functions in optimization and control problems. Set-valued Var. Anal. 18(3–4), 569–600 (2010)CrossRef MathSciNet MATH
  78.Thibault, L., Zagrodny, D.: Integration of subdifferentials of lower semicontinuous functions on Banach spaces. J. Math. Anal. Appl. 189, 33–58 (1995)CrossRef MathSciNet MATH
  79.Vial, J.-P.: Strong and weak convexity of sets and functions. Math. Oper. Research 8(2), 231–259 (1983)CrossRef MathSciNet MATH
  80.Xu, Z.-B., Roach, G.F.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157, 189–210 (1991)CrossRef MathSciNet MATH
  81.Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)CrossRef MATH
  
• 作者单位：Huynh Van Ngai (1)
  Jean-Paul Penot (2)
  
  1. Department of Mathematics, University of Quynhon, 170 An Duong Vuong, Qui Nhon, Vietnam
  2. Sorbonne Universités UPMC Université Paris 6 UMR 7598 Laboratoire Jacques-Louis Lions, 75005, Paris, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1877-0541

文摘

We study the Moreau regularization process for functions satisfying a general growth condition on general Banach spaces. We give differentiability criteria and we study the relationships between the subdifferentials of the function and the subdifferentials of its approximations. We also consider the Lasry-Lions process. Keywords Approximately convex function Convolution Monotonicity Nonsmooth analysis Regularization Subdifferential