Tangent cone

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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.)


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