Portfolio Optimization

Sec. 5.3 The Geometry of Quadratic Functions 91

To provide algebraic verification of the optimality conditions for

Example 5.2, we observe that

g(x) =



−10 + 2 x

1

−6 + 2x

2



so that g(x

0

) = (−2, −2)

′

. The cone part of the optimality conditions

requires that there is a nonnegative number u

2

with

−g(x

0

) = u

2

a

2

.

These are precisely the 2 linear equations in 1 unknown

u

2

= 2,

u

2

= 2,

which have the obvious solution u

2

= 2. Since u

2

is nonnegative, we

have algebraically verified that the optimality conditions are satisfied.

Finally, the optimal solution x

0

= (3, 2)

′

for Example 5.3 lies in the

interior of R. Since no constraint s are active at x

0

, the cone part of the

optimalit ...