Numerical Methods and Optimization

Constrained Optimization 361

x

1

x

2

x

∗

-3 -2 1 2

2

-2

FIGURE 14.3: An illustration of Example 14.4.

Convex case

Assume that the feasible set X = {x : h(x)=0,g(x) ≤ 0} is a convex set

and f(x) is a convex function over X. We will show that in this case the KKT

conditions are sufficient conditions for a global minimizer.

Theorem 14.4 Consider a convex problem

minimize f(x)

subject to h(x)=0

g(x) ≤ 0,

where all functions are continuously differentiable, the feasible set X =

{x ∈ IR

n

: h(x)=0,g(x) ≤ 0} is convex, and f (x) is convex on X.

A regular point x

∗

is a global minimizer of this problem if and only

if it satisfies the KKT conditions, that is, there exist λ ∈ IR