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2 results

  • Convergence of Subdifferentials of Convexly Composite Functions

  • C. Combari, R. Poliquin, L. Thibault
  • Journal:
    Canadian Journal of Mathematics / Volume 51 / Issue 2 / 01 April 1999
    Published online by Cambridge University Press:
    20 November 2018, pp. 250-265
    Print publication:
    01 April 1999
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  • In this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlevé-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

  • A Note on Epi-Convergence

  • Gerald Beer
  • Journal:
    Canadian Mathematical Bulletin / Volume 37 / Issue 3 / 01 September 1994
    Published online by Cambridge University Press:
    20 November 2018, pp. 294-300
    Print publication:
    01 September 1994
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  • Let LSC(X) denote the set of extended real valued lower semicontinuous functions on a metrizable space X. If f, f1, f2, f3,... is a sequence in LSC(X), we say 〈fn〉 is epi-convergent to f provided the sequence of epigraphs 〈epi fn〉 is Kuratowski- Painlevé convergent to epi f. In this note we address the following question: what conditions on f and/or on X are necessary and sufficient for this mode of convergence to force epigraphical convergence with respect to the stronger Hausdorff metric and Vietoris topologies?