Portfolio Optimization

Sec. 5.4 Optimality Conditions for QPs 101

optimality conditions for (5.16) are

A

1

x

0

≤ b

1

, A

2

x

0

= b

2

,

−∇f(x

0

) = A

′

1

u + A

′

2

v, u ≥ 0,

u

′

(A

1

x

0

− b

1

) = 0







. (5.21)

The key result here is that the multipliers associated with equality

constraints are unconstrained in sign and there is no complementary

slackness condition associated with equality constraints.

Summarizing our results, we have the following version of Theorem

5.1:

Theorem 5.2 x

0

is an optimal solution for

min{c

′

x +

1

2

x

′

Cx | A

1

x ≤ b

1

, A

2

x = b

2

}

if and only if

1. A

1

x

0

≤ b

1

, A

2

x

0

= b

2

,

2. −∇f(x

0

) = A

′

1

u + A

′

2

v, u ≥ 0,

3. u

′

(A

1

x

0

− b

1

) = 0.

The following result gives conditions under which an optimal solu-

tion ...