Paratingent cone

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

Definition

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

1. Let some ${\displaystyle {\bar {x}}\in \operatorname {cl} (S)}$ be given. An element ${\displaystyle h\in X}$ is called a tangent to ${\displaystyle S}$ at ${\displaystyle {\bar {x}}}$, if there is a sequence ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ of elements ${\displaystyle x_{n}\in S}$ and a sequence ${\displaystyle (\lambda _{n})_{n\in \mathbb {N} }}$ of positive real numbers ${\displaystyle \lambda _{n}>0}$ so that ${\displaystyle {\bar {x}}=\lim _{n\to \infty }x_{n}}$ and ${\displaystyle h=\lim _{n\to \infty }\lambda _{n}(x_{n}-{\bar {x}}).}$
2. The set ${\displaystyle T(S,{\bar {x}})}$ of all tangents to ${\displaystyle S}$ at ${\displaystyle {\bar {x}}}$ is called the contingent cone (or the Bouligand tangent cone) to ${\displaystyle S}$ at ${\displaystyle {\bar {x}}}$.[1]

References

1. Jahn, Johannes. Vector Optimization Theory, Applications, and Extensions, Second Edition, pp 90-91