Lagrange multiplier theorem

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Let LaTeX: f and LaTeX: h be smooth functions and suppose LaTeX: \nabla h(x^*) has full row rank. Then, LaTeX: x^* \in \arg\max \{f(x): h(x) = 0\} only if there exists LaTeX: v \in \mathbb{R}^m such that:

LaTeX: \nabla f(x*) - v^T \nabla h(x*) = 0.

The LaTeX: i-th component of LaTeX: v, LaTeX: v_i, is called the Lagrange multiplier associated with the LaTeX: i-th constraint, LaTeX: h_i(x)=0. Extensions to remove the rank condition and/or allow inequality constraints were by Carathéodory, John, Karush, and Kuhn & Tucker. Also see the Lagrange Multiplier Rule.

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