Portfolio Optimization

144 Portfolio Optimization with Constraints Chap. 7

or, more simply

u

0

= (0, 0, 0, −0.0055 + t 1.0673)

′

(7.12)

Notice that in (7.11), both (x

0

)

2

and (x

0

)

3

are increasing f unctions

of t so that as t is increased, their associated nonnegativity constraints

will never become active. However, (x

0

)

1

is a decreasing function of t

and is reduced to zero when

t = t

1

= 0.5455/1.7273 = 0.3158. (7.13)

Defining t

0

= 0, we have shown that x

0

(t) is optimal for this example

with multiplier vector u

0

(t) for all t with t

0

≤ t ≤ t

1

. Note that the

only active constraint in this interval is the budget constraint. It is an

equality constraint so that the optimality conditions imp ...