Tangent plane

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Consider the surface, LaTeX: \textstyle S = \{x \mbox{ in } R^n : h(x)=0\}, where LaTeX: \textstyle h \in C^1. A differentiable curve passing thru LaTeX: \textstyle x^* \in S is LaTeX: \textstyle \{x(t): x(0)=x* \mbox{ and } h(x(t))=0 \mbox{ for all } t \in (-e,e)\}, for which the derivative, LaTeX: x'(t), exists, where LaTeX: e > 0. The tangent plane at LaTeX: x^* is the set of all initial derivatives: LaTeX: \{x'(0)\}. (This is a misnomer, except in the special case of one function and two variables at a non-stationary point.) An important fact that underlies the classical Lagrange multiplier theorem when the rank of LaTeX: \textstyle \nabla h(x^*) is full row (LaTeX: x^* is then called a regular point): the tangent plane is LaTeX: \textstyle \{d: \nabla h(x^*) d = 0\}.

Extending this to allow inequalities, the equivalent of the tangent plane for a regular point LaTeX: (x^*) is the set of directions that satisfy first-order conditions to be feasible:

LaTeX: \{d: \nabla h(x*) d = 0 \mbox{ and } \nabla g_{i(x*)} d \le 0 \mbox{ for all } i: g_{i(x*)} = 0 \}.

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