LAGRANGE'S MULTIPLIER THEOREM

Lagrange's multiplier theorem - Let f, h be in C^1 and suppose grad_h(x*) has full row rank. Then, x* is in argmax(f(x): h(x)=0) only if there exists v in R^m such that:

v_i is called the Lagrange multiplier associated with the i-th constraint (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 .