# Lagrange multiplier rule

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.