The Mathematics of Financial Modelingand Investment Management

7-Optimization Page 203 Wednesday, February 4, 2004 12:50 PM

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Optimization 203

Given a multivariate function f(x 1 ,...,xn), consider the matrix

formed by the second order partial derivatives. This matrix is called the

Hessian matrix and its determinant, denoted by H, is called the Hessian

determinant (see Chapter 5 for definition of matrix and determinants):

∂

2

f ∂

2

f

--------- ··· -------------------

∂x^2 ∂x^1 ∂x

1 n

· · ·

H = · ·

·

∂^2 f

·

· ·

∂

2

f

··· ---------

∂x

∂x^2

1 ∂xn n

A point (a 1 ,...,an) is called a relative local maxima or a relative local

minima of the function fif the relationship

fa( 1 + h 1 , ..., xn+ hn ) ≤ fa( 1 , ..., an ) , h≤ d> 0

or, respectively,

fa( 1 + h 1 , ..., xn+ hn ) ≥ fa( 1 , ..., an ) , h≤ d> 0

holds for any real positive number d> 0.

A necessary, but not sufficient, condition for a point (x 1 ,...,xn) to be

a relative maximum or minimum is that all first order partial derivatives

evaluated at that point vanish, that is, that the following relationship

holds:

--------∂f - ∂f

grad[fx( 1 , ..., xn )] =  ...--------- = ( 0 , ..., 0 )

∂x 1 ∂xn

A point where the gradient vanishes is called a critical point.

A critical point can be a maximum, a minimum or a saddle point.

For functions of one variable, the following sufficient conditions hold:

■ If the first derivative evaluated at a point avanishes and the second

derivative evaluated at ais positive, then the point ais a (relative) min-

imum.