8.3.2
The Lagrange multiplier method converts the constrained minimization into an unconstrained minimization of the following Lagrange function of m+K variables
where 
are called Lagrange multipliers. For 
to be a
local minimum subject to the constraints, it is necessary that
be a stationary point of the Lagrange function:
If 
is a saddle point for which
then 
is a local minimum of 
satisfying 
[Gottfried 1973].
The following dynamics can be used to find such a saddle point
It performs energy descent on f but ascent on 
. Convergence
results of this system have been obtained by [Arrow et al. 1958]. The dynamics is recently used for neural computing [Platt and Barr 1988].
The penalty terms in the penalty function (8.52) can be added to the Lagrange function (8.55) to give an augmented Lagrange function [Powell 1969,Hestenes 1969]
where 
are finite weights. The addition of the penalty
terms do not alter the stationary point and sometimes help damp
oscillations and improve convergence. Because 
are finite, the
ill-conditioning of the problem in the penalty method when
is alleviated. After the penalty term is added, the
dynamics for 
is
