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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces
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by Ivar Ekeland and Gérard Lebourg PDF
Trans. Amer. Math. Soc. 224 (1976), 193-216 Request permission


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 $\left \| {v - u} \right \| \leqslant \eta \Rightarrow F(v) \geqslant F(u) + \langle {u^\ast },v - u\rangle - \varepsilon \left \| {v - u} \right \|$. 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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 224 (1976), 193-216
  • MSC: Primary 58C20; Secondary 49B50, 46G05
  • DOI:
  • MathSciNet review: 0431253