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  .

  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

edit
  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.