Lagrangian

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For the mathematical program

LaTeX: \min \{ f(x) : g(x) \ge 0, \, h(x) = 0, \, x \in X \},
the Lagrangian is the function:

LaTeX: L(x, u, v) = f(x) - u^T g(x) - v^T h(x).

for LaTeX: x \in X and LaTeX: u \ge 0. Note that the Lagrange Multiplier Rule can be written as the first-order conditions for LaTeX: (x^*, u^*,v^*) to be a saddle point of LaTeX: L. In Lagrange's multiplier theorem (where LaTeX: X=\mathbb{R}^n and LaTeX: g is vacuous), this is simply that LaTeX: \nabla L(x^*,v^*)=0, which could be any type of stationary point.


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