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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces

Authors: Ivar Ekeland and Gérard Lebourg
Journal: Trans. Amer. Math. Soc. 224 (1976), 193-216
MSC: Primary 58C20; Secondary 49B50, 46G05
MathSciNet review: 0431253
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Abstract: We define a function F on a Banach space V to be locally $ \varepsilon $-supported by $ {u^\ast} \in {V^\ast}$ at $ u \in V$ if there exists an $ \eta > 0$ such that $ \left\Vert {v - u} \right\Vert \leqslant \eta \Rightarrow F(v) \geqslant F(u) + \langle {u^\ast},v - u\rangle - \varepsilon \left\Vert {v - u} \right\Vert$. We prove that if the Banach space V admits a nonnegative Fréchet-differentiable function with bounded nonempty support, then, for any $ > 0$ and every lower semicontinuous function F, there is a dense set of points $ u \in V$ at which F is locally $ \varepsilon $-supported. The applications are twofold. First, to the study of functions defined as pointwise infima; we prove for instance that every concave continuous function defined on a Banach space with Fréchet-differentiable norm is Fréchet-differentiable generically (i.e. on a countable intersection of open dense subsets). Then, to the study of optimization problems depending on a parameter $ u \in V$; we give general conditions, mainly in the framework of uniformly convex Banach spaces with uniformly convex dual, under which such problems generically have a single optimal solution, depending continuously on the parameter and satisfying a first-order necessary condition.

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Article copyright: © Copyright 1976 American Mathematical Society