Portfolio Optimization

Sec. 1.2 Nonlinear Optimization 11

small neighborhood of x

0

and with (1.14) this implies f(x

0

) ≤ f(x)

for all x with Ax = b in a small neighborhood of x

0

; i.e., x

0

is a local

minimizer. In addition, if H(x) is positive definite for all x with Ax = b,

(1.14) implies that f(x

0

) ≤ f (x) for all x with Ax = b and in this case

x

0

is a global minimizer. 

The reader may recall the elementa r y Calculus formulae f or the

product and ratio of two functions:

d(uv) = vdu + udv and d

h

u

v

i

=

vdu −udv

v

2

.

We next formulate generalizations of these two rules for functions of n

va r ia bles.

Theorem 1.6 Let f (x) and g(x) be con tinuously differentiable func-

tions of the n−