Lagrange multiplier rule

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From the extension of Lagrange's multiplier theorem. Suppose

LaTeX: x^* \in \arg\max \{f(x): g(x) \le 0, \, h(x) = 0\},

where LaTeX: f, LaTeX: g, and LaTeX: h are smooth. Then, there exist multipliers LaTeX: (u, v) for which the following conditions hold:

  • LaTeX: \nabla_x [f(x^*) - u^T g(x^*) - v^T h(x^*)] = 0;
  • LaTeX: u >= 0;
  • LaTeX: u^T g(x*) = 0.

Since LaTeX: g(x^*) \le 0, the last condition, given LaTeX: u \ge 0, is equivalent to complementary slackness. These are considered first-order optimality conditions, though the Lagrange Multiplier Rule is not always valid -- see constraint qualifications.

For extensions see the Generalized Lagrange multiplier method.

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