LAGRANGIAN

Lagrangian - For the mathematical program in standard form , the Lagrangian is the function:

Note that the Lagrange Multiplier Rule can be written as the first-order conditions for (x*, u*,v*) to be a saddlepoint of L. In Lagrange's multiplier theorem (where X=R^n and g is vacuous), this is simply that grad_L(x*,v*)=0, which could be any type of stationary point . A more general Lagrangian can have nonlinear functions, U(g) and V(h), instead of the linear forms, ug and vh, respectively. In this form, U is monotonic non-decreasing (u >= 0 in the linear case), and the extension of complementary slackness is that U(g(x)) = U(0). This is a type of generalized penalty function .