Numerical Methods and Optimization

356 Numerical Methods and Optimization: An Introduction

Convex case

Consider a convex problem with equality constraints,

minimize f(x)

subject to h(x)=0,

where f(x) is a convex function and X = {x ∈ IR

n

: h(x)=0} is a convex set.

We will show that the Lagrange theorem provides sufficient conditions for a

global minimizer in this case.

Theorem 14.2 Let x

∗

be a regular point satisfying the Lagrange theo-

rem,

h(x

∗

) = 0 (14.3)

and

∇f(x

∗

)+

m



i=1

λ

i

∇h

i

(x

∗

)=0. (14.4)

Then x

∗

is a global minimizer.

Proof. From the first-order characterization of a convex function, we have

f(x) − f (x

∗

) ≥∇f (x

∗

)

T

(x − x

∗

), ∀x ∈ X. (14.5)

From the FONC (14.4),

∇f(x

∗

)=−

m



i=1

λ

i

∇h

i

(x

∗

).

So, from ...