Portfolio Optimization

96 Quadratic Programming Geometry Chap. 5

The level set for f(x) = −186.5 is thus

y

2

1

4

+

y

2

2

9

= 1 .

This is an ellipse with intercepts at (±2, 0)

′

and (0, ±3)

′

, and is shown

in Figure 5.7. Similarly, the level set for f(x) = −206.5 is

y

2

1

4

+

y

2

2

9

=

4

9

,

which is “ par allel” to the previous one (see Figure 5.7).

5.4 Optimality Conditi ons for QPs

In Section 5.2, we saw for some two-dimensional examples that an

optimal solution was characterized as being a feasible point for which

the negative gradient of the o bj ective function lies in the cone spanned

by the g r adients of the active constraints. In this section, we g eneralize

this condition to problems having ...