Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Approximation and regularization of Lipschitz functions: Convergence of the gradients

Authors: Marc-Olivier Czarnecki and Ludovic Rifford
Journal: Trans. Amer. Math. Soc. 358 (2006), 4467-4520
MSC (2000): Primary 49J45, 49J52, 57R12
Published electronically: May 9, 2006
MathSciNet review: 2231385
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We examine the possible extensions to the Lipschitzian setting of the classical result on $ C^1$-convergence: first (approximation), if a sequence $ (f_n)$ of functions of class $ C^1$ from $ \mathbb{R}^N$ to $ \mathbb{R}$ converges uniformly to a function $ f$ of class $ C^1$, then the gradient of $ f$ is a limit of gradients of $ f_n$ in the sense that $ \operatorname{graph}(\nabla f)\subset \liminf_{n\to +\infty} \operatorname{graph}(\nabla f_n)$; second (regularization), the functions $ (f_n)$ can be chosen to be of class $ C^{\infty}$ and $ C^1$-converging to $ f$ in the sense that $ \lim_{n\to +\infty} \Vert f_n-f\Vert _{\infty}+ \Vert\nabla f_n-\nabla f\Vert _{\infty}=0$. In other words, the space of $ C^{\infty}$ functions is dense in the space of $ C^1$ functions endowed with the $ C^1$ pseudo-norm.

We first deepen the properties of Warga's counterexample (1981) for the extension of the approximation part to the Lipschitzian setting. This part cannot be extended, even if one restricts the approximation schemes to the classical convolution and the Lasry-Lions regularization. We thus make more precise various results in the literature on the convergence of subdifferentials.

We then show that the regularization part can be extended to the Lipschitzian setting, namely if $ f:\mathbb{R}^N \rightarrow {\mathbb{R}}$ is a locally Lipschitz function, we build a sequence of smooth functions $ (f_n)_{n \in \mathbb{N}}$ such that

    $\displaystyle \lim_{n\to +\infty} \Vert f_n-f\Vert _{\infty}=0,$  
    $\displaystyle \lim_{n\to +\infty} d_{Haus}(\operatorname{graph}(\nabla f_n), \operatorname{graph}(\partial f))=0.$  

In other words, the space of $ C^{\infty}$ functions is dense in the space of locally Lipschitz functions endowed with an appropriate Lipschitz pseudo-distance. Up to now, Rockafellar and Wets (1998) have shown that the convolution procedure permits us to have the equality $ \limsup_{n\to +\infty} \operatorname{graph}(\nabla f_n) =\operatorname{graph}(\partial f)$, which cannot provide the exactness of our result.

As a consequence, we obtain a similar result on the regularization of epi-Lipschitz sets. With both functional and set parts, we improve previous results in the literature on the regularization of functions and sets.

References [Enhancements On Off] (What's this?)

  • 1. 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 MR 0473929 (57:13587)
  • 2. Attouch, Hédy, Variational convergence for functions and operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984 MR 0773850 (86f:49002)
  • 3. Attouch, Hédy and Beer, Gerald, On the convergence of subdifferentials of convex functions. Arch. Math. (Basel) 60 (1993), no. 4, 389-400 MR 1206324 (94b:49018)
  • 4. Aubin, Jean-Pierre and Frankowska, Halina, Set-valued analysis. Systems and control: foundations and applications, Birkhäuser, 1990 MR 1048347 (91d:49001)
  • 5. 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 MR 1187139 (94c:46082)
  • 6. Benoist, Joël, Approximation and regularization of arbitrary sets in finite dimension, Set Valued Anal. 2 (1994), no. 1-2, pp. 95-115 MR 1285823 (95e:49019)
  • 7. Bourbaki, Nicolas, Eléments de mathématique, Hermann, Paris
  • 8. 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 MR 0709590 (85m:49002), MR 1019086 (90g:49011), MR 1058436 (91e:49001)
  • 9. Clarke, Frank. H.; Ledyaev, Yuri S.; Stern, Ronald J.; Wolenski, Peter R., Nonsmooth analysis and control theory. Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998 MR 1488695 (99a:49001)
  • 10. Clarke, Frank. H.; Stern, Ronald. J.; Wolenski, Peter R., Proximal smoothness and the lower-$ C\sp 2$ property. J. Convex Anal. 2 (1995), no. 1-2, 117-144 MR 1363364 (96j:49014)
  • 11. Cornet, Bernard and Czarnecki, Marc-Olivier, Smooth representations of epi-Lipschitz subsets of $ \mathbb{R}^ n$. Nonlinear Anal. 37 (1999), no. 2, Ser. A: Theory Methods, 139-160 MR 1689740 (2000a:49031)
  • 12. Cornet, Bernard and Czarnecki, Marc-Olivier, Smooth normal approximations of epi-Lipschitz subsets of $ R\sp n$. SIAM J. Control Optim. 37 (1999), no. 3, 710-730 MR 1675157 (2000b:49031)
  • 13. Cornet, Bernard and Czarnecki, Marc-Olivier, Existence of (generalized) equilibria: necessary and sufficient conditions. Comm. Appl. Nonlinear Anal. 7 (2000), no. 1, 21-53 MR 1733400 (2000j:49029)
  • 14. Cornet, Bernard and Czarnecki, Marc-Olivier, Existence of generalized equilibria. Nonlinear Anal. 44 (2001), no. 5, Ser. A: Theory Methods, 555-574 MR 1822230 (2001m:90131)
  • 15. Cornet, Bernard and Czarnecki, Marc-Olivier, Smoothing the distance function to a closed subset of $ \mathbb{R}^{n}$ and applications, manuscript (1998)
  • 16. Czarnecki, Marc-Olivier, thèse de doctorat, Université Paris 1 Panthéon Sorbonne, 1996
  • 17. Federer, Herbert, Curvature measures. Trans. Amer. Math. Soc. 93 (1959) 418-491 MR 0110078 (22:961)
  • 18. 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 MR 1775860 (2001h:49025)
  • 19. Ioffe, Alexander. D., Approximate subdifferentials and applications. I. The finite-dimensional theory. Trans. Amer. Math. Soc. 281 (1984), no. 1, 389-416 MR 0719677 (84m:49029)
  • 20. Jourani, Abderrahim, Limit superior of subdifferentials of uniformly convergent functions. Positivity 3 (1999), no. 1, 33-47 MR 1675463 (2000i:49022)
  • 21. 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
  • 22. Kuratowski, Casimir, Topologie. Vol. II. (French) 2ème éd. Monografie Matematyczne, Tom XXI. Polskie Towarzystwo Matematyczne, Warszawa, 1952 MR 0054232 (14:892a)
  • 23. Lasry, Jean-Michel. and Lions, Pierre-Lions, A remark on regularization in Hilbert spaces. Israel J. Math. 55 (1986), no. 3, 257-266 MR 0876394 (88b:41020)
  • 24. 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 MR 0535804 (80d:49004)
  • 25. 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 MR 1290725 (95k:49035)
  • 26. 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); translated from Prikl. Mat. Meh. 40 (1976), no. 6, 1014-1023 (Russian) MR 0487669 (58:7284)
  • 27. Mordukhovich, Boris. S., Approximation methods in problems of optimization and control, Nauka, Moscow, 1988 MR 0945143 (89m:49001)
  • 28. Plaskacz, S\lawomir, On the solution sets for differential inclusions. Boll. Un. Mat. Ital. A (7) 6 (1992), no. 3, 387-394 MR 1196133 (93m:34021)
  • 29. 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 MR 1145732 (93a:49013)
  • 30. Poliquin, René. A.; Rockafellar, R. Tyrrell., Prox-regular functions in variational analysis. Trans. Amer. Math. Soc. 348 (1996), no. 5, 1805-1838 MR 1333397 (96h:49039)
  • 31. Poliquin, René. A.; Rockafellar, R. Tyrrell.; Thibault, Lionel, Local differentiability of distance functions. Trans. Amer. Math. Soc. 352 (2000), no. 11, 5231-5249 MR 1694378 (2001b:49024)
  • 32. Rademacher, Hans, Uber partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale, Math. Ann. 79 (1919), 340-359 MR 1511935
  • 33. Rockafellar, R. Tyrrell, Clarke's tangent cones and the boundaries of closed sets in $ R\sp{n}$. Nonlinear Anal. 3 (1979), no. 1, 145-154 (1978) MR 0520481 (80d:49032)
  • 34. Rockafellar, R. Tyrrell, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization. Math. Oper. Res. 6 (1981), no. 3, 424-436 MR 0629642 (83m:90088)
  • 35. Rockafellar, R. Tyrrell and Wets, Roger J.-B., Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317. Springer-Verlag, Berlin, 1998 MR 1491362 (98m:49001)
  • 36. Subbotin, Andrei I., Generalized solutions of first order PDEs, Birkhäuser, Boston, 1995 MR 1320507 (96b:49002)
  • 37. Warga, Jack, Fat homeomorphisms and unbounded derivate containers. J. Math. Anal. Appl. 81 (1981), no. 2, 545-560 MR 0622836 (83f:58007)
  • 38. Zolezzi, Tullio, Convergence of generalized gradients. Set convergence in nonlinear analysis and optimization. Set-Valued Anal. 2 (1994), no. 1-2, 381-393 MR 1285841 (95f:49016)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 49J45, 49J52, 57R12

Retrieve articles in all journals with MSC (2000): 49J45, 49J52, 57R12

Additional Information

Marc-Olivier Czarnecki
Affiliation: Institut de Mathematiques et Modelisation de Montpellier UMR 5030 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier Cedex 5, France

Ludovic Rifford
Affiliation: Institut Girard Desargues, Université Claude Bernard Lyon I, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
Address at time of publication: Departement de Mathématiques (Bâtiment 425), Université de Paris-Sud, 91405 Orsay, France

Keywords: Lipschitz functions, epi-Lipschitz sets, approximation, regularization, gradient convergence, normal cone
Received by editor(s): January 28, 2003
Received by editor(s) in revised form: August 28, 2004
Published electronically: May 9, 2006
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society