Portfolio Optimization

6 Optimization Chap. 1

In this format, a

i

is the gradient of the i −th constraint function.

In Example 1.1, the key condition was that ∇f( x) be a multiple

of the gradient of the constraint. The generalization of this for (1.4) is

that

∇f(x) = u

1

a

1

+ u

2

a

2

+ ··· + u

m

a

m

,

which says that ∇f(x) must be a linear combination of the gradients

of the constraints. By replacing u

i

with −u

i

, this condition becomes

− ∇f(x) = u

1

a

1

+ u

2

a

2

+ ··· + u

m

a

m

. (1.5)

Later, when we deal with optimality conditions for problems with in-

equality constraints it will be natural to f ormulate them in terms of

−∇f(x) rather than ∇f(x). To be compatible with those conditions, we

will form ...