Helping You Finding an Appropriate Regularization Process

Article

Abstract

We survey some processes that relate a given function to a more regular function. We examine the compensated convexity process from this point of view and we give a special attention to an infimal convolution approximation generalizing the Moreau approximation which can be applied to nonconvex functions satisfying mild growth conditions.

Keywords

Compensated convexity Convolution Differentiability Infimal convolution Nonsmooth analysis Regularization Subdifferential 

Mathematics Subject Classification (2010)

49J52 46N10 46T20 

Notes

Acknowledgements

We would like to thank the three referees for their careful reading of the initial version of our manuscript and for their insightful comments.

References

  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel (2008)MATHGoogle Scholar
  2. 2.
    Asplund, E.: Fréchet differentiability of convex functions. Acta Math. 121, 31–47 (1968)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Attouch, H.: Variational Convergence of Functions and Operators. Pitman, London (1984)MATHGoogle Scholar
  4. 4.
    Attouch, H., Azé, D.: Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry–Lions method. Ann. Inst. Henri Poincaré, (C) 10, 289–312 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhaüser, Boston (1990)MATHGoogle Scholar
  6. 6.
    Bačák, M., Borwein, J.M., Eberhard, A., Mordukhovich, B.: Infimal convolutions and Lipschitzian properties of subdifferentials for prox-regular functions in Hilbert spaces. J. Convex Anal. 17, 737–763 (2010)MathSciNetMATHGoogle Scholar
  7. 7.
    Beauzamy, B.: Introduction to Banach Spaces and Their Geometry. Mathematics Studies, vol. 68. North-Holland, Amsterdam (1982)MATHGoogle Scholar
  8. 8.
    Beck, A., Teboulle, M.: Smoothing and first order methods: a unified framework. SIAM J. Optim. 22, 557–580 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Benoist, J.: Convergence de la dérivée de la régularisée de Lasry-Lions. C. R. Acad. Sci. Paris 315, 941–944 (1992)MathSciNetMATHGoogle Scholar
  10. 10.
    Benoist, J.: Approximation and regularization of arbitrary sets in finite dimensions. Set-Valued Anal. 2, 95–115 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Benyamini, Y, Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. Colloquium Publications, vol. 48. American Mathematical Society, Providence (2000)MATHGoogle Scholar
  12. 12.
    Bernard, F., Thibault, L.: Prox-regular functions in Hilbert spaces. J. Math. Anal. Appl. 303, 1–14 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bernard, F., Thibault, L.: Uniform prox-regularity of functions and epigraphs in Hilbert spaces. Nonlinear Anal. 60, 187–207 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Bernard, F., Thibault, L., Zlateva, N.: Characterizations of prox-regular sets in uniformly convex Banach spaces. J. Convex Anal. 13, 525–559 (2006)MathSciNetMATHGoogle Scholar
  15. 15.
    Bernard, F., Thibault, L., Zlateva, N.: Prox-regular sets and epigraphs in uniformly convex Banach spaces: various regularities and other properties. Trans. Am. Math. Soc. 363, 2211–2247 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Bernard, F., Thibault, L., Zagrodny, D.: Integration of primal lower nice functions in Hilbert spaces. J. Optim. Theory Appl. 124, 561–579 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Borwein, J.M., Giles, J.R.: The proximal normal formula in Banach space. Trans. Am. Math. Soc. 302, 371–381 (1987)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. CMS Books in Mathematics, vol. 20. Springer, New York (2005)Google Scholar
  19. 19.
    Bougeard, M., Penot, J.-P., Pommellet, A.: Towards minimal assumptions for the infimal convolution regularization. J. Approx. Theory 64, 245–270 (1991)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)MATHGoogle Scholar
  21. 21.
    Burke, J.V., Hoheisel, T.: Epi-convergent smoothing with applications to convex composite functions. SIAM J. Optim. 23, 1457–1479 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Cabot, A., Jourani, A., Thibault, L.: Envelopes for sets and functions: regularization and generalized conjugacy. Mathematika 63, 383–432 (2017)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Birkhäuser, Boston (2004)MATHGoogle Scholar
  24. 24.
    Cepedello-Boiso, M.: Approximation of Lipschitz functions by Δ-convex functions in Banach spaces. Isr. J. Math. 106, 269–284 (1998)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Cepedello Boiso, M.: On regularization in super-reflexive Banach spaces by infimal convolution formulas. Stud. Math. 129, 265–284 (1998)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Clarke, F.H.: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol. 264. Springer-Verlag, London (2013)CrossRefGoogle Scholar
  27. 27.
    Colombo, G., Thibault, L.: Prox-regular sets and applications. In: Gao, D. Y., Motreanu, D (eds.) Handbook of Nonconvex Analysis and Applications, pp 99–182. International Press, Somerville (2010)Google Scholar
  28. 28.
    De Giorgi, E., Marino, A., Tosques, M.: Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8) 68, 180–187 (1980)MathSciNetMATHGoogle Scholar
  29. 29.
    Diestel, J.: Geometry of Banach Spaces—Select Topics. Lecture Notes in Mathematics, vol. 485. Springer-Verlag, Berlin Heidelberg (1975)CrossRefGoogle Scholar
  30. 30.
    Fabián, M.J.: Sub differentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss. Acta Univ. Carol. Math. Phys. 30, 51–56 (1989)MATHGoogle Scholar
  31. 31.
    Hájek, P., Johanis, M.: Smooth approximations. J. Funct. Anal. 259, 561–582 (2010)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Ioffe, A.: Euler-Lagrange and Hamiltonian formalisms in dynamic optimization. Trans. Am. Math. Soc. 349, 2871–2900 (1997)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Johanis, M.: Approximation of Lipschitz mappings. Serdica Math. J. 29, 141–148 (2003)MathSciNetMATHGoogle Scholar
  34. 34.
    Jourani, A., Thibault, L., Zagrodny, D.: Differential properties of the Moreau envelope. J. Funct. Anal. 266, 1185–1237 (2014)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kecis, I., Thibault, L.: Subdifferential characterization of s-lower regular function. Appl. Anal. 94, 85–98 (2015)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Kecis, I., Thibault, L.: Moreau envelopes of s-lower regular functions. Nonlinear Anal. 127, 157–181 (2015)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Lasry, J.-M., Lions, P.-L.: A remark on regularization in Hilbert spaces. Isr. J. Math. 55, 257–266 (1986)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Lemaréchal, C., Sagastizábal, C.: Practical aspects of the Moreau–Yosida regularization: theoretical preliminaries. SIAM J. Optim. 7, 367–385 (1997)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Levy, A.B., Poliquin, R.A., Thibault, L.: Partial extensions of Attouch’s theorem with applications to proto-derivatives of subgradient mappings. Trans. Am. Math. Soc. 347, 1269–1294 (1995)MathSciNetMATHGoogle Scholar
  40. 40.
    Martínez-Legaz, J.-E., Penot, J.-P.: Regularization by erasement. Math. Scand. 98, 97–124 (2006)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Mazade, M., Thibault, L.: Differential variational inequalities with locally prox-regular sets. J. Convex Anal. 19, 1109–1139 (2012)MathSciNetMATHGoogle Scholar
  42. 42.
    Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 348, 1235–1280 (1996)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I. Basic Theory. Grundlehren der Mathematischen Wissenschaften, vol. 330. Springer-Verlag, Berlin Heidelberg (2006)CrossRefGoogle Scholar
  44. 44.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. II. Applications. Grundlehren der Mathematischen Wissenschaften, vol. 331. Springer-Verlag, Berlin Heidelberg (2006)CrossRefGoogle Scholar
  45. 45.
    Ngai, H.V., Théra, M.: Phi-regular functions in Asplund spaces. Control Cybern. 36, 755–774 (2007)MATHGoogle Scholar
  46. 46.
    Ngai, H.V., Penot, J.-P.: Approximately convex functions and approximately monotonic operators. Nonlinear Anal. 66, 547–564 (2007)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Ngai, H.V., Penot, J.-P.: Paraconvex functions and paraconvex sets. Stud. Math. 184, 1–29 (2008)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Ngai, H.V., Penot, J.-P.: Approximately convex sets. J. Nonlinear Convex Anal. 8, 337–371 (2007)MathSciNetMATHGoogle Scholar
  49. 49.
    Ngai, H.V., Penot, J.-P.: Subdifferentiation of regularized functions. Set-Valued Var. Anal. 24, 167–189 (2016)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Ngai, H.V., Penot, J.-P.: Paraconvex regularization. in preparationGoogle Scholar
  51. 51.
    Penot, J.-P., Bougeard, M.: Approximation and decomposition properties of some classes of locally d.c. functions. Math. Program. 41, 195–227 (1988)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Penot, J.-P., Bougeard, M.L.: Approximation and decomposition properties of some classes of locally D.C. functions. Math. Program. 41, 195–227 (1988)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Penot, J.-P.: Proximal mappings. J. Approx. Theory 94, 203–221 (1998)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Penot, J.-P.: Calculus Without Derivatives. Graduate Texts in Mathematics, vol. 266. Springer, New York (2013)CrossRefGoogle Scholar
  55. 55.
    Penot, J.-P.: Analysis: From Concepts to Applications. Universitext. Springer, London (2016)CrossRefGoogle Scholar
  56. 56.
    Penot, J.-P., Ratsimahalo, R.: On the Yosida approximation of operators. Proc. R. Soc. Edinb. Sect. A 131, 945–966 (2001)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Poliquin, R.A.: Integration of subdifferentials of nonconvex functions. Nonlinear Anal. 17, 385–398 (1991)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Poliquin, R.A.: An extension of Attouch’s theorem and its application to second-order epi-differentiation of convexly composite functions. Trans. Am. Math. Soc. 322, 861–874 (1992)MathSciNetMATHGoogle Scholar
  59. 59.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren der Mathematishen Wissenschaften, vol. 317. Springer-Verlag, Berlin Heidelberg (2002)Google Scholar
  60. 60.
    Rolewicz, S.: On the coincidence of some subdifferentials in the class of α(⋅)-paraconvex functions. Optimization 50, 353–360 (2001)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Seeger, R.: Smoothing nondifferentiable convex functions: the technique of the rolling ball. Rev. Mat. Apl. 18, 45–60 (1997)MathSciNetMATHGoogle Scholar
  62. 62.
    Thibault, L., Zagrodny, D.: Integration of subdifferentials of lower semicontinuous functions on Banach spaces. J. Math. Anal. Appl. 189, 33–58 (1995)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Vial, J.-P.: Strong and weak convexity of sets and functions. Math. Oper. Res. 8, 231–259 (1983)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Xu, Z.-B., Roach, G.F.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157, 189–210 (1991)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)CrossRefMATHGoogle Scholar
  66. 66.
    Zhang, K.: On various semiconvex relaxations of the squared-distance function. Proc. R. Soc. Edinb. Sect. A 129, 1309–1323 (1999)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Zhang, K.: Compensated convexity and its applications. Ann. Inst. H. Poincaré, (C) Non Linéaire Anal. 25, 743–771 (2008)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Zhang, K., Crooks, E., Orlando, A.: Compensated convexity, multiscale medial axis maps and sharp regularity of the squared-distance function. SIAM J. Math. Anal. 47, 4289–4331 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Quy NhonQui NhonVietnam
  2. 2.Sorbonne Universités UPMC Université Paris 6 UMR 7598 Laboratoire Jacques-Louis LionsParisFrance

Personalised recommendations