In mathematics, the paratingent cone and contingent cone were introduced by Bouligand (1932), and are closely related to tangent cones.
Definition
[edit]Let be a nonempty subset of a real normed vector space .
- Let some be a point in the closure of . An element is called a tangent (or tangent vector) to at , if there is a sequence of elements and a sequence of positive real numbers such that and
- The set of all tangents to at is called the contingent cone (or the Bouligand tangent cone) to at .[1]
An equivalent definition is given in terms of a distance function and the limit infimum. As before, let be a normed vector space and take some nonempty set . For each , let the distance function to be
Then, the contingent cone to at is defined by[2]
References
[edit]- ^ Johannes, Jahn (2011). Vector Optimization. Springer Berlin Heidelberg. pp. 90–91. doi:10.1007/978-3-642-17005-8. ISBN 978-3-642-17005-8.
- ^ Aubin, Jean-Pierre; Frankowska, Hèléne (2009). "Chapter 4: Tangent Cones". Set-Valued Analysis. Modern Birkhäuser Classics. Boston: Birkhäuser. p. 121. doi:10.1007/978-0-8176-4848-0_4. ISBN 978-0-8176-4848-0.
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