Set-Valued Analysis

, Volume 1, Issue 4, pp 379–392

Second-order epi-derivatives of integral functionals

  • A. B. Levy

DOI: 10.1007/BF01027827

Cite this article as:
Levy, A.B. Set-Valued Anal (1993) 1: 379. doi:10.1007/BF01027827


Epi-derivatives have many applications in optimization as approached through nonsmooth analysis. In particular, second-order epi-derivatives can be used to obtain optimality conditions and carry out sensitivity analysis. Therefore the existence of second-order epi-derivatives for various classes of functions is a topic of considerable interest. A broad class of composite functions on ℝn called ‘fully amenable’ functions (which include general penalty functions composed withC2 mappings, possibly under a constraint qualification) are now known to be twice epi-differentiable. Integral functionals appear widely in problems in infinite-dimensional optimization, yet to date, only integral functionals defined by convex integrands have been shown to be twice epi-differentiable, provided that the integrands are twice epi-differentiable. Here it is shown that integral functionals are twice epi-differentiable even without convexity, provided only that their defining integrands are twice epi-differentiable and satisfy a uniform lower boundedness condition. In particular, integral functionals defined by fully amenable integrands are twice epi-differentiable under mild conditions on the behavior of the integrands.

Mathematics Subject Classifications (1991)

Primary: 49J52 Secondary: 58C20 

Key words

Generalized second derivatives nonsmooth analysis epi-derivatives fully amenable functions integral functionals optimization 

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • A. B. Levy
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleU.S.A.

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