LAGRANGE MULTIPLIER RULE

Lagrange Multiplier Rule (LMR). From the extension to Lagrange's multiplier theorem :

Suppose x* is in argmax(f(x): g(x) <= 0, h(x) = 0), where f, g, h are in C^1. Then, there exist multipliers (u, v) for which the following conditions hold:

1. grad_x_[f(x*) - ug(x*) - vh(x*)] = 0;
2. u >= 0;
3. ug(x*)=0.

Since g(x*) <= 0, the last condition, given u >= 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 .