# Tangent cone

Let $LaTeX: \textstyle \mbox{S} \subset R^n$ and let $LaTeX: \textstyle x^* \in S.$ The tangent cone, $LaTeX: \textstyle \mbox{T(S}, x*),$ is the set of points that have the following limit properties: $LaTeX: \textstyle y$ is in $LaTeX: \textstyle \mbox{T(S}, x*)$ if there exist sequences $LaTeX: \textstyle a_n \ge 0 \mbox{ and } x^n \in S$ such that $LaTeX: \textstyle \{x^n\} \to x^* \mbox{ and } \{||a_n (x^n - x^*) - y||\} \to 0.$ This arises in connection with the Lagrange Multiplier Rule much like the tangent plane, though it allows for more general constraints – e.g., set constraints. In particular, when there are only equality constraints, $LaTeX: \textstyle h(x) = 0, \mbox{ T(S}, x*) = \mbox{null space }$ of $LaTeX: \textstyle \nabla h(x^*)$ if $LaTeX: \textstyle \nabla h(x^*)$ has full row rank. (There are some subtleties that render the tangent cone more general, in some sense, than the tangent plane or null space. It is used in establishing a necessary constraint qualification.)