Compactness and Boundedness for a Class of Concave-Convex Functions

  • E. Cavazzuti
  • N. Pacchiarotti
Part of the Ettore Majorana International Science Series book series (EMISS, volume 43)


In this chapter we generalize some results (see for instance [1]), which have been proved for sequences of convex functions, to saddle functions. More specifically, we are concerned here with concave-convex functions defined on finite dimensional spaces with values in the extended real line: the main result is Compactness Theorem 4.2, which allows us to give sufficient conditions in order to obtain the closure of epi-hypo limits by means of the pointwise limits of Yosida approximates.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H. Attouch. “Variational convergence for functions and operator”. Pitmann A.P.P. (1984).Google Scholar
  2. [2]
    H. Attouch and R.B. Wets. “A convergence theory for saddle functions”. Trans. Amer. Math. Soc., vol. 280 1 (1983), pp. 1–41.CrossRefGoogle Scholar
  3. [3]
    E. Cavazzuti. “F-convergenze multiple: convergenza di punti di sella e di max-min”. Boll. Un. Mat. Ital., (6) 1-B (1982), pp. 251–274.Google Scholar
  4. [4]
    E. Cavazzuti. “Convergence of equilibria in the theory of games”. In “Optimization and Related Fields”, R. Conti et al. (eds.). Lecture Notes in Math, No. 1190, Springer-Verlag (1986), pp.95–130.Google Scholar
  5. [5]
    E. Cavazzuti. “Compactness for concave functions”. (To appear).Google Scholar
  6. [6]
    T. Franzoni. “Abstract F-convergence”. In “Optimization and Related Fields”, R. Conti et al. (Eds.). Lecture Notes in Math. No. 1190, Springer-Verlag (1986), pp. 229–242.Google Scholar
  7. [7]
    G.H. Greco. “Minimax theorems and saddling transformations”. (To appear).Google Scholar
  8. G.H. Greco. “Decomposizioni di semifiltri e F-limiti sequenziali in reticoli cornpletamente distributivi”. Ann. Mat.Pura e Applicata. Serie IV. T. 137 (1984), pp. 61–81.Google Scholar
  9. [9]
    L. McLinden. “A minmax theorem”. Math. Op. Res. (9), 4 (1984), pp. 576591.Google Scholar
  10. [10]
    R.T. Rockafellar. “Convex analysis”. Princeton University Press (1970).Google Scholar

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • E. Cavazzuti
    • 1
  • N. Pacchiarotti
    • 2
  1. 1.Dept. of MathematicsUniv. of ModenaModenaItaly
  2. 2.Dept. of MathematicsUniv. of PadovaPadovaItaly

Personalised recommendations