Abstract
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 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.
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This work was supported in part by the National Science Foundation under grant DMS-9200303.
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Levy, A.B. Second-order epi-derivatives of integral functionals. Set-Valued Anal 1, 379–392 (1993). https://doi.org/10.1007/BF01027827
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DOI: https://doi.org/10.1007/BF01027827