Skip to main content
SpringerLink
Log in
Book cover

Ill-posed Variational Problems and Regularization Techniques pp 203–227Cite as

Some Links Between Approximation, Nonsmooth Analysis, Penalization and Regularization

  • Conference paper

  • 225 Accesses

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 477))

Abstract

We point out some connections between the different subjects of the title. We also present a simple approach to the viscosity selection principle of H. Attouch which avoids the use of epi-convergence.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Aussel, J.-N. Corvellec and M. Lassonde, Nonsmooth constrained optimization and multidirectional mean value inequalities, SIAM J. Optim., to appear.

    Google Scholar 

  2. H. Attouch, Viscosity solutions of minimization problems, epi-convergence and scaling, Séminaire d’Analyse Convexe, Montpellier, vol 22 (1992), 8.1-8.48.

    Google Scholar 

  3. H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim. 6(3) (1996), 769–806.

    Article  Google Scholar 

  4. H. Attouch and D. Azé, Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method, Ann. Institut H. Poincaré Anal, non linéaire 10(3) (1993), 289–312.

    Google Scholar 

  5. D. Azé and A. Rahmouni, Intrinsic bounds for Kuhn-Tucker points of perturbed convex programs, in “ Recent developments in optimization, Seventh French-German Conference on Optimization”, R. Durier and C. Michelot (eds), Lecture Notes in Economics and Math. Systems # 429 Springer Verlag, Berlin (1995), 17–35.

    Google Scholar 

  6. V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Ed. Academiei Rep. socialiste romania (Bucharest)-Noordhoff (Leyden) (1976).

    Google Scholar 

  7. E.N. Barron and W. Liu, Calculus of variations in L∞, Applied Math. Opt. 35 (1997), 237–243.

    Google Scholar 

  8. E.N. Barron, R. Jensen and W. Liu, Hopf-Lax formula for u t+ H(u, Du) = 0. J. Differ. Eq. 126 (1996), 48–61.

    Article  Google Scholar 

  9. J.M. Borwein, J. S. Treiman and Q.J. Zhu, Necessary conditions for constrained optimization problems with semicontinuous and continuous data, Trans. Amer. Math. Soc. 350 (1998), 2409–2429.

    Article  Google Scholar 

  10. J.M. Borwein, Q.J. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control and Optim. 34(5) (1996), 1568–1591.

    Article  Google Scholar 

  11. J.M. Borwein and Q.J. Zhu, A survey of subdifferential calculus with applications, to appear in J. Nonlinear Anal. Th. Meth. Appl.

    Google Scholar 

  12. M. Bougeard, J.-P. Penot, Approximation and decomposition properties of some classes of locally d. c. functions, Math. Program. 41 (1988), 195–227.

    Article  Google Scholar 

  13. M. Bougeard, J.-P. Penot and A. Pommellet, Towards minimal assumptions for the infimal convolution regularization, J. Approx. Th. 64(3) (1991), 245–270.

    Article  Google Scholar 

  14. H. Brézis, Opérateurs maximaux monotones, North Holland, Amsterdam, 1973.

    Google Scholar 

  15. F.E. Browder, Nonlinear variational inequalities and maximal monotone mappings in Banach spaces, Math. Ann. 183 (1969), 213–231.

    Article  Google Scholar 

  16. G. Chavent, Strategies for the regularization of nonlinear least squares problems, in “Inverse problems in diffusion processes”, H.W. Engl and W. Rundell (eds.), SIAM-GAMM, SIAM Philadelphia (1995), 217-232.

    Google Scholar 

  17. G. Chavent and K. Kunish, Convergence of Tikhonov regularization for constrained ill-posed inverse problems, Inverse Problems 10 (1994), 63–76.

    Article  Google Scholar 

  18. G. Chavent and K. Kunish, On weakly nonlinear inverse problems, SIAM J. Appl Math. 56(2) (1996), 542–572.

    Article  Google Scholar 

  19. F.H. Clarke and Yu. S. Ledyaev, Mean value inequalities in Hilbert spaces, Trans. Amer. Math. Soc. 344 (1994), 307–324.

    Article  Google Scholar 

  20. F.H. Clarke, Yu. S. Ledyaev, R.J. Stern, Complements, approximations, smoothings and invariance properties, J. Convex Anal. 4(2) (1997), 189–219.

    Google Scholar 

  21. F.H. Clarke, R.J. Stern, P.R. Wolenski, Proximal smoothness and the lower-C 2 property, J. Convex Anal. 2 (1995), 117–144.

    Google Scholar 

  22. A. Gioan, Regularized minimization under weaker hypotheses, Appl Math. Optim. 8 (1981), 59–67.

    Article  Google Scholar 

  23. C.W. Groetsch, The theory of Tychonov regularization for Fredholm equations of the first kind, Research Notes in Maths 105, Pitman, Boston, 1984.

    Google Scholar 

  24. A.D. Ioffe, Fuzzy principles and characterization of trustworthiness, Set-Valued Anal. 6 (3) (1998), to appear.

    Google Scholar 

  25. G. Isac and M. Théra, Complementary problem and the existence of postcritical equilibrium state of the thin elastic plate, Séminaire d’Analyse numérique, Université Paul Sabatier, Toulouse III (1985–1986), XI-1-XI-27.

    Google Scholar 

  26. A. Kaplan and R. Tichatschke, Stable methods for ill-posed variational problems. Prox-regularization of elliptic variational inequalities and semiinfinite optimization problems, Akademie Verlag, Berlin, 1994.

    Google Scholar 

  27. A. Kaplan and R. Tichatschke, Proximal point methods and nonconvex optimization, J. Global Opt. 13 (4) (1998).

    Google Scholar 

  28. M. Lassonde, First order rules for nonsmooth constrained optimization, preprint, Univ. Antilles-Guyane, Pointe-à-Pitre, 1998

    Google Scholar 

  29. J.-P. Penot, A characterization of tangential regularity, Nonlinear Anal. Th. Meth. Appl. 25(6) (1981), 625–643.

    Article  Google Scholar 

  30. J.-P. Penot, Compact nets, filters and relations, J. Math. Anal Appl. 93(2) (1983), 400–417.

    Article  Google Scholar 

  31. J.-P. Penot, Conditioning convex and nonconvex problems, J. Optim. Th. Appl. 90(3) (1995), 539–548.

    Google Scholar 

  32. J.-P. Penot, Are generalized derivatives useful for generalized convex functions? in “Generalized convexity, generalized monotonicity: recent results”, J.-P. Crouzeix et al (eds), Kluwer, Dordrecht (1998), 3-59.

    Google Scholar 

  33. J.-P. Penot, Proximal mappings, J. Approx. Theory, 94 (1998), 203–221.

    Article  Google Scholar 

  34. J.-P. Penot, Well-behavior, well-posedness and nonsmooth analysis, Pliska Stud. Math. Bulgar. 12 (1998), 1001–1050.

    Google Scholar 

  35. J.-P. Penot, What is quasiconvex analysis?, Optimization, to appear.

    Google Scholar 

  36. J.-P. Penot and R. Ratsimahalo, On the Yosida approximation of operators, preprint, Univ. of Pau, 1997, revised July 1998.

    Google Scholar 

  37. J.-P. Penot and C. Zălinescu, Harmonic sum and duality, preprint, Univ. of Pau, 1999.

    Google Scholar 

  38. J.-P. Penot and C. Zălinescu, Elements of quasiconvex subdifferential calculus, preprint (1998), J. Convex Anal., to appear.

    Google Scholar 

  39. J.-P. Penot and C. Zălinescu, Regularization of quasiconvex functions, in preparation.

    Google Scholar 

  40. F. Plastria, Lower subdifferentiable functions and their minimization by cutting plane, J. Optim. Th. Appl. 46(1) (1994), 37–54.

    Article  Google Scholar 

  41. R.T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, N.J., 1970.

    Google Scholar 

  42. R.T. Rockafellar and R. J-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1997.

    Google Scholar 

  43. A. Seeger, Direct and inverse addition in convex analysis and applications, J. Math. Anal. Appl. 148 (1990), 317–341.

    Article  Google Scholar 

  44. A. Seeger, M. Volle, On a convolution operation obtained by adding level sets: classical and new results, Recherche Opérationnelle 29 (1995), 131–154.

    Google Scholar 

  45. A. Tikhonov et V. Arsénine, Méthodes de résolution de problèmes mal posés, Mir, Moscow, (1976).

    Google Scholar 

  46. S. Traore and M. Volle, Quasi-convex conjugation and Mosco convergence, Richerche di Mat. 44(2) (1995), 369–388.

    Google Scholar 

  47. S. Traore and M. Volle, Dualité pour la minimization du maximum de deux fonctions convexes: applications à la somme en niveaux des fonctions convexes, preprint, Univ. of Avignon, (1996).

    Google Scholar 

  48. S. Traoré and M. Volle, Epiconvergence d’une suite de sommes en niveaux de fonctions convexes, preprint, Univ. of Avignon, (1996).

    Google Scholar 

  49. M. Volle, Duality for the level sum of quasiconvex functions and applications, ESAIM: Control, Optimisation and Calculus of Variations, 3 (1998), 329–343, http://www.emath.fr/cocv/

    Article  Google Scholar 

  50. C. Zălinescu, On an abstract control problem, Numer. Fund. Anal. Appl. 2(6) (1980), 531–542.

    Article  Google Scholar 

  51. C. Zălinescu, Stability for a class of nonlinear optimization problems and applications, in “Nonsmooth Optimization and Related Topics”, F.H. Clarke, V.F. Dem’yanov and F. Giannessi, eds, Plenum Press, New York, (1989), 437–458.

    Google Scholar 

  52. C. Zălinescu, A note on d-stability of convex programs and limiting Lagrangians, Math. Progr. 53 (1992), 267–277.

    Article  Google Scholar 

  53. C. Zälinescu, Mathematical programming in infinite dimensional normed linear spaces (Rumanian) Editura Academiei, Bucharest (1998), French translation to appear.

    Google Scholar 

  54. E. Zeidler, Nonlinear Functional Analysis and its Applications, Springer Verlag, Berlin, (1990).

    Book  Google Scholar 

  55. Q.J. Zhu, The equivalence of several basic theorems for subdifferentials, Set-Valued Anal. 6(2) (1998), 171–185.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathématiques Appliquées, UPRES CNRS, 5033, Av. de l’Université, 64000, Pau, France

    Jean-Paul Penot

Authors
  1. Jean-Paul Penot
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LACO, URA 1586, University of Limoges, 123 Avenue Albert Thomas, 87060, Limoges Cedex, France

    Michel Théra

  2. Fachbereich IV — Mathematik, University of Trier, 54286, Trier, Germany

    Rainer Tichatschke

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Penot, JP. (1999). Some Links Between Approximation, Nonsmooth Analysis, Penalization and Regularization. In: Théra, M., Tichatschke, R. (eds) Ill-posed Variational Problems and Regularization Techniques. Lecture Notes in Economics and Mathematical Systems, vol 477. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45780-7_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-45780-7_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66323-2

  • Online ISBN: 978-3-642-45780-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation