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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Convex subcones of the contingent cone in nonsmooth calculus and optimization


Author: Doug Ward
Journal: Trans. Amer. Math. Soc. 302 (1987), 661-682
MSC: Primary 58C20; Secondary 46G05, 90C30
Corrigendum: Trans. Amer. Math. Soc. 311 (1989), 429-431.
MathSciNet review: 891640
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Abstract: The tangential approximants most useful in nonsmooth analysis and optimization are those which lie "between" the Clarke tangent cone and the Bouligand contigent cone. A study of this class of tangent cones is undertaken here. It is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone, there are such convex subcones which are more "accurate" approximants than the Clarke tangent cone and possess an associated subdifferential calculus that is equally strong. In addition, a large class of convex subcones of the contingent cone can replace the Clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program. However, the Clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven. Overall, the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0891640-2
PII: S 0002-9947(1987)0891640-2
Keywords: Clarke tangent cone, contingent cone, subgradient, Liusternik theorem, strong general position
Article copyright: © Copyright 1987 American Mathematical Society