Editing Tangent cone (section)

== Definitions in nonlinear analysis ==
In nonlinear analysis, there are many definitions for a tangent cone, including the [[adjacent cone]], [[Georges Bouligand|Bouligand]]'s [[contingent cone]], and the [[Clarke tangent cone]]. These three cones coincide for a convex set, but they can differ on more general sets.

=== Clarke tangent cone ===
Let <math>A</math> be a nonempty closed subset of the [[Banach space]] <math>X</math>. The Clarke's tangent cone to <math>A</math> at <math>x_0\in A</math>, denoted by <math>\widehat{T}_A(x_0)</math> consists of all vectors <math>v\in X</math>, such that for any sequence <math>\{t_n\}_{n\ge 1}\subset\mathbb{R}</math> tending to zero, and any sequence <math>\{x_n\}_{n\ge 1}\subset A</math> tending to <math>x_0</math>, there exists a sequence <math>\{v_n\}_{n\ge 1}\subset X</math> tending to <math>v</math>, such that for all <math>n\ge 1</math> holds <math>x_n+t_nv_n\in A</math>

Clarke's tangent cone is always subset of the corresponding [[contingent cone]] (and coincides with it, when the set in question is convex). It has the important property of being a closed convex cone.