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Volumn 358, Issue 10, 2006, Pages 4467-4520

Approximation and regularization of Lipschitz functions: Convergence of the gradients

Author keywords

Approximation; Epi Lipschitz sets; Gradient convergence; Lipschitz functions; Normal cone; Regularization

Indexed keywords


EID: 33749509031     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-06-04103-1     Document Type: Article
Times cited : (42)

References (39)
  • 1
    • 0342889716 scopus 로고
    • Convergence de fonctions convexes, des sous-différentiels et semi-groupes associés
    • (French) A539-A542 MR0473929 (57:13587)
    • Attouch, Hédy, Convergence de fonctions convexes, des sous-différentiels et semi-groupes associés. (French) C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 10, A539-A542 MR0473929 (57:13587)
    • (1977) C. R. Acad. Sci. Paris Sér. A-B , vol.284 , Issue.10
    • Attouch, H.1
  • 2
    • 0003513119 scopus 로고
    • Variational convergence for functions and operators
    • Pitman (Advanced Publishing Program), Boston, MA, MR0773850 (86f:49002)
    • Attouch, Hédy, Variational convergence for functions and operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984 MR0773850 (86f:49002)
    • (1984) Applicable Mathematics Series
    • Attouch, H.1
  • 3
    • 0002469251 scopus 로고
    • On the convergence of subdifferentials of convex functions
    • MR1206324 (94b:49018)
    • Attouch, Hédy and Beer, Gerald, On the convergence of subdifferentials of convex functions. Arch. Math. (Basel) 60 (1993), no. 4, 389-400 MR1206324 (94b:49018)
    • (1993) Arch. Math. (Basel) , vol.60 , Issue.4 , pp. 389-400
    • Attouch, H.1    Beer, G.2
  • 5
    • 0000010928 scopus 로고
    • Convergence de la dérivée de la régularisée Lasry-Lions
    • (French) MR1187139 (94c:46082)
    • Benoist, Joël, Convergence de la dérivée de la régularisée Lasry-Lions. (French) [Convergence of the derivative of the Lasry-Lions regularization] C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 8, 941-944 MR1187139 (94c:46082)
    • (1992) C. R. Acad. Sci. Paris Sér. I Math. , vol.315 , Issue.8 , pp. 941-944
    • Benoist, J.1
  • 6
    • 0010942514 scopus 로고
    • Approximation and regularization of arbitrary sets in finite dimension
    • MR1285823 (95e:49019)
    • Benoist, Joël, Approximation and regularization of arbitrary sets in finite dimension, Set Valued Anal. 2 (1994), no. 1-2, pp. 95-115 MR1285823 (95e:49019)
    • (1994) Set Valued Anal. , vol.2 , Issue.1-2 , pp. 95-115
    • Benoist, J.1
  • 8
    • 0003859933 scopus 로고
    • Optimization and nonsmooth analysis
    • John Wiley & Sons, Inc., New York, Reprint of the 1983 original. Université de Montréal, Centre de Recherches Mathèmatiques, Montréal, QC, 1989. Second edition. Classics in Applied Mathematics, 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990 MR0709590 (85m:49002), MR1019086 (90g:49011), MR1058436 (91e:49001)
    • Clarke, Frank H., Optimization and nonsmooth analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1983. Reprint of the 1983 original. Université de Montréal, Centre de Recherches Mathèmatiques, Montréal, QC, 1989. Second edition. Classics in Applied Mathematics, 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990 MR0709590 (85m:49002), MR1019086 (90g:49011), MR1058436 (91e:49001)
    • (1983) Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication
    • Clarke Frank, H.1
  • 11
    • 0032607121 scopus 로고    scopus 로고
    • n
    • Ser. A: Theory Methods, 139-160 MR1689740 (2000a:49031)
    • n. Nonlinear Anal. 37 (1999), no. 2, Ser. A: Theory Methods, 139-160 MR1689740 (2000a:49031)
    • (1999) Nonlinear Anal. , vol.37 , Issue.2
    • Cornet, B.1    Czarnecki, M.-O.2
  • 13
    • 0141697020 scopus 로고    scopus 로고
    • Existence of (generalized) equilibria: Necessary and sufficient conditions
    • MR1733400 (2000j:49029)
    • Cornet, Bernard and Czarnecki, Marc-Olivier, Existence of (generalized) equilibria: necessary and sufficient conditions. Comm. Appl. Nonlinear Anal. 7 (2000), no. 1, 21-53 MR1733400 (2000j:49029)
    • (2000) Comm. Appl. Nonlinear Anal. , vol.7 , Issue.1 , pp. 21-53
    • Cornet, B.1    Czarnecki, M.-O.2
  • 14
    • 0035341612 scopus 로고    scopus 로고
    • Existence of generalized equilibria
    • Ser. A: Theory Methods, 555-574 MR1822230 (2001m:90131)
    • Cornet, Bernard and Czarnecki, Marc-Olivier, Existence of generalized equilibria. Nonlinear Anal. 44 (2001), no. 5, Ser. A: Theory Methods, 555-574 MR1822230 (2001m:90131)
    • (2001) Nonlinear Anal. , vol.44 , Issue.5
    • Cornet, B.1    Czarnecki, M.-O.2
  • 16
    • 33749528583 scopus 로고    scopus 로고
    • thèse de doctorat, Université Paris 1 Panthéon Sorbonne
    • Czarnecki, Marc-Olivier, thèse de doctorat, Université Paris 1 Panthéon Sorbonne, 1996
    • (1996)
    • Czarnecki, M.-O.1
  • 17
    • 84968469617 scopus 로고
    • Curvature measures
    • MR0110078 (22:961)
    • Federer, Herbert, Curvature measures. Trans. Amer. Math. Soc. 93 (1959) 418-491 MR0110078 (22:961)
    • (1959) Trans. Amer. Math. Soc. , vol.93 , pp. 418-491
    • Federer, H.1
  • 18
    • 0034662561 scopus 로고    scopus 로고
    • Reconstruction of the Clarke subdifferential by the Lasry-Lions regularizations
    • MR1775860 (2001h:49025)
    • Georgiev, Pando Gr. and Zlateva, Nadia P., Reconstruction of the Clarke subdifferential by the Lasry-Lions regularizations. J. Math. Anal. Appl. 248 (2000), no. 2, 415-428 MR1775860 (2001h:49025)
    • (2000) J. Math. Anal. Appl. , vol.248 , Issue.2 , pp. 415-428
    • Georgiev, P.Gr.1    Zlateva Nadia, P.2
  • 19
    • 84967717450 scopus 로고
    • Approximate subdifferentials and applications. I. The finite-dimensional theory
    • MR0719677 (84m:49029)
    • Ioffe, Alexander. D., Approximate subdifferentials and applications. I. The finite-dimensional theory. Trans. Amer. Math. Soc. 281 (1984), no. 1, 389-416 MR0719677 (84m:49029)
    • (1984) Trans. Amer. Math. Soc. , vol.281 , Issue.1 , pp. 389-416
    • Ioffe Alexander, D.1
  • 20
    • 27144495510 scopus 로고    scopus 로고
    • Limit superior of subdifferentials of uniformly convergent functions
    • MR1675463 (2000i:49022)
    • Jourani, Abderrahim, Limit superior of subdifferentials of uniformly convergent functions. Positivity 3 (1999), no. 1, 33-47 MR1675463 (2000i:49022)
    • (1999) Positivity , vol.3 , Issue.1 , pp. 33-47
    • Jourani, A.1
  • 21
    • 0042584067 scopus 로고
    • Generalized normals and derivatives, and necessary optimality conditions in nondifferentiable programming
    • (in Russian), Moscow
    • Kruger, A. Ya. and Mordukhovich, Boris. S., Generalized normals and derivatives, and necessary optimality conditions in nondifferentiable programming (in Russian), Depon. VINITI 494-80, Moscow, 1980
    • (1980) Depon. VINITI 494-80
    • Kruger, A.Ya.1    Mordukhovich, B.S.2
  • 22
    • 33749526741 scopus 로고
    • (French) 2ème éd. Monografie Matematyczne, Tom XXI. Polskie Towarzystwo Matematyczne, Warszawa, MR0054232 (14:892a)
    • Kuratowski, Casimir, Topologie. Vol. II. (French) 2ème éd. Monografie Matematyczne, Tom XXI. Polskie Towarzystwo Matematyczne, Warszawa, 1952 MR0054232 (14:892a)
    • (1952) Topologie , vol.2
    • Kuratowski, C.1
  • 23
    • 51249173208 scopus 로고
    • A remark on regularization in Hilbert spaces
    • MR0876394 (88b:41020)
    • Lasry, Jean-Michel. and Lions, Pierre-Lions, A remark on regularization in Hilbert spaces. Israel J. Math. 55 (1986), no. 3, 257-266 MR0876394 (88b:41020)
    • (1986) Israel J. Math. , vol.55 , Issue.3 , pp. 257-266
    • Lasry, J.-M.1    Lions, P.-L.2
  • 24
    • 33749521914 scopus 로고
    • Sous-dérivabilité de fonctions semi-continues et convergence de dérivées: Quelques résultats en densité
    • (French) MR0535804 (80d:49004)
    • Lebourg, Gérard, Sous-dérivabilité de fonctions semi-continues et convergence de dérivées: quelques résultats en densité. (French) C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 16, A753-A755 MR0535804 (80d:49004)
    • (1979) C. R. Acad. Sci. Paris Sér. A-B , vol.288 , Issue.16
    • Lebourg, G.1
  • 25
    • 84966232303 scopus 로고
    • Thibault, Lionel, Partial extensions of Attouch's theorem with applications to proto-derivatives of subgradient mappings
    • MR1290725 (95k:49035)
    • Levy, A. B.; Poliquin, René A; Thibault, Lionel, Partial extensions of Attouch's theorem with applications to proto-derivatives of subgradient mappings. Trans. Amer. Math. Soc. 347 (1995), no. 4, 1269-1294 MR1290725 (95k:49035)
    • (1995) Trans. Amer. Math. Soc. , vol.347 , Issue.4 , pp. 1269-1294
    • Levy, A.B.1    Poliquin René, A.2
  • 26
    • 33646940121 scopus 로고
    • Maximum principle in the problem of time optimal response with nonsmooth constraints
    • Mordukhovich, Boris. Sh., Maximum principle in the problem of time optimal response with nonsmooth constraints. J. Appl. Math. Mech. 40 (1976), no. 6, 960-969 (1977);
    • (1976) J. Appl. Math. Mech. , vol.40 , Issue.6 , pp. 960-969
    • Mordukhovich, B.Sh.1
  • 27
    • 0008830845 scopus 로고
    • translated from (Russian) MR0487669 (58:7284)
    • translated from Prikl. Mat. Meh. 40 (1976), no. 6, 1014-1023 (Russian) MR0487669 (58:7284)
    • (1976) Prikl. Mat. Meh. , vol.40 , Issue.6 , pp. 1014-1023
  • 29
    • 0000704354 scopus 로고
    • On the solution sets for differential inclusions
    • MR1196133 (93m:34021)
    • Plaskacz, Sławomir, On the solution sets for differential inclusions. Boll. Un. Mat. Ital. A (7) 6 (1992), no. 3, 387-394 MR1196133 (93m:34021)
    • (1992) Boll. Un. Mat. Ital. A , vol.6 , Issue.3-7 , pp. 387-394
    • Plaskacz, S.1
  • 30
    • 84968500669 scopus 로고
    • An extension of Attouch's theorem and its application to second-order epi-differentiation of convexly composite functions
    • MR1145732 (93a:49013)
    • Poliquin, René A., An extension of Attouch's theorem and its application to second-order epi-differentiation of convexly composite functions. Trans. Amer. Math. Soc. 332 (1992), no. 2, 861-874 MR1145732 (93a:49013)
    • (1992) Trans. Amer. Math. Soc. , vol.332 , Issue.2 , pp. 861-874
    • Poliquin, R.A.1
  • 31
    • 0007633319 scopus 로고    scopus 로고
    • Prox-regular functions in variational analysis
    • MR1333397 (96h:49039)
    • Poliquin, René. A.; Rockafellar, R. Tyrrell., Prox-regular functions in variational analysis. Trans. Amer. Math. Soc. 348 (1996), no. 5, 1805-1838 MR1333397 (96h:49039)
    • (1996) Trans. Amer. Math. Soc. , vol.348 , Issue.5 , pp. 1805-1838
    • Poliquin, R.A.1    Rockafellar, R.T.2
  • 32
    • 23044523810 scopus 로고    scopus 로고
    • Local differentiability of distance functions
    • MR1694378 (2001b:49024)
    • Poliquin, René. A.; Rockafellar, R. Tyrrell.; Thibault, Lionel, Local differentiability of distance functions. Trans. Amer. Math. Soc. 352 (2000), no. 11, 5231-5249 MR1694378 (2001b:49024)
    • (2000) Trans. Amer. Math. Soc. , vol.352 , Issue.11 , pp. 5231-5249
    • Poliquin, R.A.1    Rockafellar, R.T.2    Thibault, L.3
  • 33
    • 0003176679 scopus 로고
    • Uber partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale
    • MR1511935
    • Rademacher, Hans, Uber partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale, Math. Ann. 79 (1919), 340-359 MR1511935
    • (1919) Math. Ann. , vol.79 , pp. 340-359
    • Rademacher, H.1
  • 34
    • 0000670389 scopus 로고
    • n
    • MR0520481 (80d:49032)
    • n. Nonlinear Anal. 3 (1979), no. 1, 145-154 (1978) MR0520481 (80d:49032)
    • (1978) Nonlinear Anal. , vol.3 , Issue.1 , pp. 145-154
    • Rockafellar, R.T.1
  • 35
    • 0000752102 scopus 로고
    • Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization
    • MR0629642 (83m:90088)
    • Rockafellar, R. Tyrrell, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization. Math. Oper. Res. 6 (1981), no. 3, 424-436 MR0629642 (83m:90088)
    • (1981) Math. Oper. Res. , vol.6 , Issue.3 , pp. 424-436
    • Rockafellar, R.T.1
  • 38
    • 0000169893 scopus 로고
    • Fat homeomorphisms and unbounded derivate containers
    • MR0622836 (83f:58007)
    • Warga, Jack, Fat homeomorphisms and unbounded derivate containers. J. Math. Anal. Appl. 81 (1981), no. 2, 545-560 MR0622836 (83f:58007)
    • (1981) J. Math. Anal. Appl. , vol.81 , Issue.2 , pp. 545-560
    • Warga, J.1
  • 39
    • 0012305580 scopus 로고
    • Convergence of generalized gradients. Set convergence in nonlinear analysis and optimization
    • MR1285841 (95f:49016)
    • Zolezzi, Tullio, Convergence of generalized gradients. Set convergence in nonlinear analysis and optimization. Set-Valued Anal. 2 (1994), no. 1-2, 381-393 MR1285841 (95f:49016)
    • (1994) Set-Valued Anal. , vol.2 , Issue.1-2 , pp. 381-393
    • Zolezzi, T.1


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