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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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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

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 $\left \| {v - u} \right \| \leqslant \eta \Rightarrow F(v) \geqslant F(u) + \langle {u^\ast },v - u\rangle - \varepsilon \left \| {v - u} \right \|$. 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.
References
  • Edgar Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31–47. MR 231199, DOI 10.1007/BF02391908
  • Edgar Asplund, Farthest points in reflexive locally uniformly rotund Banach spaces, Israel J. Math. 4 (1966), 213–216. MR 206662, DOI 10.1007/BF02771633
  • J. Baranger, Existence de solutions pour des problèmes d’optimisation non convexe, J. Math. Pures Appl. (9) 52 (1973), 377–405 (1974) (French). MR 380360
  • J. Baranger and R. Temam, Nonconvex optimization problems depending on a parameter, SIAM J. Control 13 (1975), 146–152. MR 0430901, DOI 10.1137/0313008
  • M. F. Bidaut, Théorèmes d’existence et d’existence en général d’un contrôle optimal pour des systèmes régis par des équations aux dérivées partielles non linéaires, Thèse, Université de Paris, 1973.
  • N. Bourbaki, Éléments de mathématique. Part I. Les structures fondamentales de l’analyse. Livre III. Topologie générale. Chapitres I et II, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 858, Hermann & Cie, Paris, 1940 (French). MR 0004747
  • Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
  • Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094, DOI 10.1007/BFb0082079
  • Michael Edelstein, Farthest points of sets in uniformly convex Banach spaces, Israel J. Math. 4 (1966), 171–176. MR 203426, DOI 10.1007/BF02760075
  • M. Edelstein, On nearest points of sets in uniformly convex Banach spaces, J. London Math. Soc. 43 (1968), 375–377. MR 226364, DOI 10.1112/jlms/s1-43.1.375
  • I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. MR 346619, DOI 10.1016/0022-247X(74)90025-0
  • Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Collection Études Mathématiques, Dunod, Paris; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). MR 0463993
  • K. John and V. Zizler, Smoothness and its equivalents in weakly compactly generated Banach spaces, J. Functional Analysis 15 (1974), 1–11. MR 0417759, DOI 10.1016/0022-1236(74)90021-4
  • S. B. Stečkin, Caractérisation d l’approximation par des sous-ensembles d’espaces vectoriels normés, Rev. Math. Pures Appl. 8 (1963), 5-18.
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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: https://doi.org/10.1090/S0002-9947-1976-0431253-2
  • MathSciNet review: 0431253