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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

Let S {\displaystyle S} be a nonempty subset of a real normed vector space ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} .

  1. Let some x ¯ ∈ cl ⁡ ( S ) {\displaystyle {\bar {x}}\in \operatorname {cl} (S)} be a point in the closure of S {\displaystyle S} . An element h ∈ X {\displaystyle h\in X} is called a tangent (or tangent vector) to S {\displaystyle S} at x ¯ {\displaystyle {\bar {x}}} , if there is a sequence ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} of elements x n ∈ S {\displaystyle x_{n}\in S} and a sequence ( λ n ) n ∈ N {\displaystyle (\lambda _{n})_{n\in \mathbb {N} }} of positive real numbers λ n > 0 {\displaystyle \lambda _{n}>0} such that x ¯ = lim n → ∞ x n {\displaystyle {\bar {x}}=\lim _{n\to \infty }x_{n}} and h = lim n → ∞ λ n ( x n − x ¯ ) . {\displaystyle h=\lim _{n\to \infty }\lambda _{n}(x_{n}-{\bar {x}}).}
  2. The set T ( S , x ¯ ) {\displaystyle T(S,{\bar {x}})} of all tangents to S {\displaystyle S} at x ¯ {\displaystyle {\bar {x}}} is called the contingent cone (or the Bouligand tangent cone) to S {\displaystyle S} at x ¯ {\displaystyle {\bar {x}}} .

An equivalent definition is given in terms of a distance function and the limit infimum. As before, let ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} be a normed vector space and take some nonempty set S ⊂ X {\displaystyle S\subset X} . For each x ∈ X {\displaystyle x\in X} , let the distance function to S {\displaystyle S} be

d S ( x ) := inf { ‖ x − x ′ ‖ ∣ x ′ ∈ S } . {\displaystyle d_{S}(x):=\inf\{\|x-x'\|\mid x'\in S\}.}

Then, the contingent cone to S ⊂ X {\displaystyle S\subset X} at x ∈ cl ⁡ ( S ) {\displaystyle x\in \operatorname {cl} (S)} is defined by

T S ( x ) := { v : lim inf h → 0 + d S ( x + h v ) h = 0 } . {\displaystyle T_{S}(x):=\left\{v:\liminf _{h\to 0^{+}}{\frac {d_{S}(x+hv)}{h}}=0\right\}.}

References

  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.


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