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Transactions of the American Mathematical Society

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

 
 

 

Regular points of Lipschitz functions


Author: Alexander D. Ioffe
Journal: Trans. Amer. Math. Soc. 251 (1979), 61-69
MSC: Primary 58E15; Secondary 49B99
DOI: https://doi.org/10.1090/S0002-9947-1979-0531969-6
MathSciNet review: 531969
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Abstract: Let f be a locally Lipschitz function on a Banach space X, and S a subset of X. We define regular (i.e. noncritical) points for f relative to S, and give a sufficient condition for a point $ z \, \in \, S$ to be regular. This condition is then expressed in the particular case when f is $ {C^1}$, and is used to obtain a new proof of Hoffman's inequality in linear programming.


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  • [1] N. Bourbaki, Variétiés différentielles et analytiques, Hermann, Paris, 1971. MR 0281115 (43:6834)
  • [2] F. H. Clarke, Necessary conditions for nonsmooth problems in optimal control, Ph. D. Thesis, Univ. of Washington, 1973.
  • [3] -, A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976), 165-174. MR 0414104 (54:2209)
  • [4] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. MR 0346619 (49:11344)
  • [5] A. Hoffman, On approximate solutions of systems of linear inequalities, J. Res. Nat. Bur. Standards Sect. B 49 (1952), 263-265. MR 0051275 (14:455b)
  • [6] A. D. Ioffe, Necessary and sufficient conditions for a local minimum (to appear).
  • [7] A. D. Ioffe and V. M. Tikhomirov, Theory of extremal problems, ``Nauka", Moscow, 1974. (Russian) MR 0410502 (53:14251)
  • [8] G. Lebourg, Valeur moyenne pour gradient généralisé, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), 795-797. MR 0388097 (52:8934)
  • [9] L. A. Ljusternik and V. I. Sobolev, Elements of functional analysis, ``Nauka", Moscow, 1965. MR 0209802 (35:698)
  • [10] S. M. Robinson, An application of error bounds for convex programming in a linear space, SIAM J. Control 13 (1975), 271-273. MR 0385671 (52:6531)
  • [11] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N. J., 1970.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0531969-6
Article copyright: © Copyright 1979 American Mathematical Society

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