Generalization of Lagrange Multipliers
In general, to find an extremal value associated with a n-dimensional function
given by f =f( ̄x) = f(x 1 , x 2 ,... , x n) subject to kconstraint conditions that can be
written in the form gi( ̄x) = gi(x 1 , x 2 ,... , x n) = 0, for i= 1, 2 ,.. ., k, where kis less than
n. It is required that the gradient vectors ∇g 1 ,∇g 2 ,... ,∇gkbe linearly independent
vectors, then one can employ the method of Lagrange multipliers as follows. The
Lagrangian rule requires that the function F =f+
∑ki=1λigi can be written in the
expanded form
(7 .84)which contains the objective function f, summed with each of the constraint func-
tions gi, multiplied by a Lagrange multiplier λi,for the index ihaving the values
i= 1,... , k. Here the function Fand consequently the function fhas stationary values
at those points where the following equations are satisfied
∂F∂x i=0, for i= 1,... , n
∂F
∂λ j=0, for j= 1,... , k
(7 .85)The equations (7.85) represent a system of (n+k)equations in the (n+k)unknowns
x 1 , x 2 ,... , x n, λ 1 , λ 2 ,... , λ kfor determining the stationary points. In general, the sta-
tionary points will be found in terms of the λivalues. The vector ( ̄x 0 , ̄λ 0 )where x ̄ 0 and
̄λ 0 are solutions of the system of equations (7.85) can be thought of as critical points
associated with the Lagrangian function F( ̄x, ̄λ) given by equation (7.84). The re-
sulting stationary points must then be tested to determine whether they correspond
to a relative maximum value, minimum value or saddle point. One can form the
Hessian^7 matrix associated with the function F( ̄x;λ ̄)and analyze this matrix at the
(^7) See page 318 for definition of Hessian matrix.