Set-Valued Analysis

, Volume 1, Issue 4, pp 379-392

First online:

Second-order epi-derivatives of integral functionals

  • A. B. LevyAffiliated withDepartment of Mathematics, University of Washington

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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 ℝ n called ‘fully amenable’ functions (which include general penalty functions composed withC 2 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