Portfolio Optimization

154 Portfolio Optimization with Constraints Chap. 7

It now follows that

(v

j

)

i

(t) ≥ 0 for all t with

˜

t

jl

≤ t ≤

˜

t

ju

, and i with 1 ≤ k

i

≤ m. (7.37)

Another way of stating (7.37) is that the multipliers for the active in-

equality constraints remain nonnegative for all t in the interval [

˜

t

jl

,

˜

t

ju

].

Note that we have used “l” and “u” as part of the subscripts of

˜

t

jl

and

˜

t

ju

to indicate lower and upper bounds, resp ectively.

We next turn to restrictions on t imposed by inequality constraints

which are inactive at t

∗

. Let x

j

(t) be as in ( 7.32). Then x

j

(t) must

satisfy

a

′

i

x

j

(t) ≤ b

i

for all with i 6∈ K

j

and 1 ≤ i ≤ m. (7.38)

Substitution of the expression for x