The Mathematics of Financial Modelingand Investment Management

7-Optimization Page 205 Wednesday, February 4, 2004 12:50 PM

Optimization 205

∂f ∂f 

∇f= ---------, ...,--------- = ( 0 , ..., 0 )

∂x 1 ∂xn 

Let’s now discuss how to find maxima and minima when the optimi-

zation problem has equality constraints. Suppose that the n variables

(x 1 ,...,xn) are not independent, but satisfy m < n constraint equations

g 1 (x 1 ,...,xn) = 0

.

.

.

gm(x 1 ,...,xn) = 0

These equations define, in general, an (n-m)-dimensional surface.

For instance, in the case of two variables, a constraint g 1 (x,y) = 0

defines a line. In the case of three variables, one constraint g 1 (x,y,z) = 0

defines a two-dimensional surface while two constraints g 1 (x,y,z) = 0,

g 2 (x,y,z) = 0 define a line in the three-dimensional space, and so on.

Our objective is to find the maxima or minima of the function f for

the set of points that also satisfy the constraints. It can be demonstrated

that, under this restriction, the gradient ∇f of f need not vanish at the

maxima or minima, but need only be orthogonal to the (n-m)-dimen-

sional surface described by the constraint equations. That is, the follow-

ing relationships must hold

∇f= λλλλT∇g, for some λλλλ= (λ 1 , ...λ, m )

or, in the usual notation

m

∂f ∂gj

-------- = ∑λj --------, i = 1,...,n

∂xi j = 1 ∂xi

The coefficients (λ 1 ,...,λm) are called Lagrange multipliers.

If we define the function

m

Fx( 1 , ...,xn, λ 1 , ...λ, m )= fx( 1 , ...,xn )– ∑λjgj

j = 1