Set-Valued Analysis

, Volume 4, Issue 2, pp 135–155

Cutting and scanning methods in set-valued analysis I. An epigraphical and graphical calculus

  • H. Attouch
  • M. Volle
Article
  • 32 Downloads

Abstract

Cutting analysis of sets (scanning, laser exploration, etc.), when applied to epigraphs of functions and graphs of operators, gives rise to a rich calculus and provides a unifying approach to various operations in optimization and variational analysis.

Mathematics Subject Classifications (1991)

41A65 65K10 

Key words

epigraphical analysis cutting methods scanning analysis slice sublevel sets epigraphical sum sublevel sum convexity quasi-convexity parallel sum of operators 

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References

  1. 1.
    Artstein, Z.: Distribution of random sets and random selections, Israël J. Math. 46 (1983), 313–324.Google Scholar
  2. 2.
    Attouch, H.: Variational Convergence for Functions and Operators, Applicable Maths. Series, Pitman, London, 1984.Google Scholar
  3. 3.
    Attouch, H. and Thera, M.: Convergences en analyse multivoque et unilatérale, MATAPLI, Bulletin de liaison 36 (1993), 23–40.Google Scholar
  4. 4.
    Attouch, H. and Riahi, H.: Stability results for Ekeland's ε-variational principle and cone extremal solutions, Math. Oper. Res. 18(1) (1993), 173–201.Google Scholar
  5. 5.
    Attouch, H. and Wets, R.: Epigraphical analysis, in H. Attouch, J.-P. Aubin, F. Clarke, and I. Ekeland (eds), Analyse non linéaire, Gauthier-Villars, Paris, 1989, pp. 73–100.Google Scholar
  6. 6.
    Aubin, J.-P.: Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam, 1979.Google Scholar
  7. 7.
    Aubin, J.-P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Basel, 1990.Google Scholar
  8. 8.
    Auslender, A. and Cominetti, R.: First and second order sensitivity analysis of nonlinear programs under constraint qualification condition, Optimization 21, (1990), 351–363.Google Scholar
  9. 9.
    Beer, G.: Conjugate convex functions and the epidistance topology, Proc. Amer. Math. Soc. 108 (1990), 117–128.Google Scholar
  10. 10.
    Beer, G. and Lucchetti, R.: Minima of quasi-convex functions, Optimization 20 (1989), 581–596.Google Scholar
  11. 11.
    Beer, G., Rockafellar, R.-T., and Wets, R.: A characterisation of epi-convergence in terms of convergence of level-sets, Proc. Amer. Math. Soc. 116 (1992), 753–761.Google Scholar
  12. 12.
    Borwein, J.: On the existence of Pareto Efficient Points, Math. Oper. Res. 81 (1983), 64–73.Google Scholar
  13. 13.
    Buttazzo, G.: Su una definizione generale dei Γ-limit, Boll. Un. Mat. Ital. 14-B (1977), 722–744.Google Scholar
  14. 14.
    Castaing, C. and Valadier, M.: Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, New York, 1977.Google Scholar
  15. 15.
    Clarke, F.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983.Google Scholar
  16. 16.
    Crouzeix, J.-P.: Contributions à l'étude des fonctions quasi-convexes, Thèse d'Etat, Université de Clermont, 1977.Google Scholar
  17. 17.
    Dal Maso, G.: An Introduction to Γ-Convergence, Birkhäuser, Basel, 1993.Google Scholar
  18. 18.
    De Giorgi, E.: Sulla convergenza di alcune successioni di integrali del tipo dell'area, Rend. di. Mat. 4(8) (1975), 277–294.Google Scholar
  19. 19.
    Ekeland, I.: On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.Google Scholar
  20. 20.
    Elqortobi, A.: Inf-convolution quasi-convexe des fonctionnelles positives, Rech. Opérationnelle 26 (1992), 301–311.Google Scholar
  21. 21.
    Greco, G.: Teoria dos semifiltros, Seminario Brasileiro de Analise, Rio de Janeiro, 1985.Google Scholar
  22. 22.
    Hess, C.: Contributions à l'étude de la mesurabilité, de la loi de probabilité, et de la convergence des multifonctions, Thèse, Montpellier, 1986.Google Scholar
  23. 23.
    Modica, L.: Gradient theory of phase transitions and minimal interface criteria, Arch. Rat. Mech. Anal. 98, (1987), 123–142.Google Scholar
  24. 24.
    Moreau, J.-J.: Inf-convolution, sous-additivité, convexité des fonctions numériques, J. Math. Pures Appl. 49 (1970), 109–154.Google Scholar
  25. 25.
    Passty, G.: The parallel sum of non-linear monotone operators, Nonlinear Anal. TMA 10, (1986).Google Scholar
  26. 26.
    Penot, J.-P.: Miscellaneous incidences of convergence theories in optimization, and nonlinear analysis I: Behavior of solutions, Set-Valued Analysis 2 (1994), 259–274.Google Scholar
  27. 27.
    Poliquin, R.: An extension of Attouch's theorem and its application to second order epidifferentiation of convex composite functions, Trans. Amer. Math. Soc. 332 (1992), 861–874.Google Scholar
  28. 28.
    Rockafellar, R. T.: Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1970.Google Scholar
  29. 29.
    Rockafellar, R. T.: First and second order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), 75–108.Google Scholar
  30. 30.
    Rockafellar, R. T. and Wets, R.: Set-valued analysis and subdifferential calculus, in preparation.Google Scholar
  31. 31.
    Seeger, A.: Analyse du second ordre de problèmes non différentiables, Thèse Université P. Sabatier, 1986.Google Scholar
  32. 32.
    Seeger, A. and Volle, M.: On a convolution operator obtained by adding level sets: classical and new results, Oper. Res. 29(2) (1995), 131–154.Google Scholar
  33. 33.
    Truffert, A.: Conditional expectations of integrands and random sets, Ann. Oper. Res. 30 (1991), 117–156.Google Scholar
  34. 34.
    Volle, M.: Convergence en niveaux et en épigraphes, C.R. Acad. Sci. Paris, Série I 299 (1984), 295–298.Google Scholar
  35. 35.
    Volle, M.: Quelques résultats relatifs à l'approche par les tranches de l'épi-convergence, Contributions à la dualité en optimisation et à l'épi-convergence, Thèse d'état, Université de Pau, 1986.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • H. Attouch
    • 1
  • M. Volle
    • 2
  1. 1.Département des Sciences mathématiquesUniversité Montpellier 2Montpellier Cedex 5France
  2. 2.Département de MathématiquesUniversité d'AvignonAvignonFrance

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