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The Quantificational Tangent Cones

Published online by Cambridge University Press:  20 November 2018

Doug Ward
Doug Ward*
Affiliation:
Miami University, Oxford, Ohio
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Nonsmooth analysis has provided important new mathematical tools for the study of problems in optimization and other areas of analysis [1, 2, 6-12, 28]. The basic building blocks of this subject are local approximations to sets called tangent cones.

Definition 1.1. Let E be a real, locally convex, Hausdorff topological vector space (abbreviated l.c.s.). A tangent cone (on E) is a mapping A:2E × E → 2E such that A(C, x) is a (possibly empty) cone for all nonempty C in 2E and x in E.

In the sequel, we will say that a tangent cone has a certain property (e.g. “A is closed” or “A is convex“) if A(C, x) has that property for all non-empty sets C and all x in C. (If A(C, x) is empty, it will be counted as having the property trivially.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 3 , 01 June 1988 , pp. 666 - 694
Copyright
Copyright © Canadian Mathematical Society 1988

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