Portfolio Optimization

Sec. 6.1 QPSolver: A QP Solution Algorithm 11 7

For the situation in (6.8), we would have

−∇f(x

1

) = (−(v

0

)

1

)a

5

+ (−(v

0

)

2

)a

10

+ (−(v

0

)

3

)a

15

,

If this problem had m = 20 constraints, we could define the multiplier

vector u by

u

i

= 0, i = 1 , 2, . . . , 20, i 6= 5, 10, 1 5,

u

5

= −(v

0

)

1

, u

10

= −(v

0

)

2

, u

15

= −(v

0

)

3

.

Thus, the point x

1

together with the multiplier vector u, satisfies all of

the optimality conditions with the possible exception of the nonnega-

tivity requirement for u.

Returning t o the general situation (6.16), we can construct a multi-

plier vector u from −v

0

in a similar manner. If u ≥ 0, then x

1

together

with u satisfies all of the optimality conditions ...