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Paratingent cone

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In mathematics, the paratingent cone and contingent cone were introduced by Bouligand (1932), and are closely related to tangent cones.

Definition

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Let be a nonempty subset of a real normed vector space .

  1. 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
  2. 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

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  1. ^ 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.
  2. ^ 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.