Unconstrained Minimization
Assume:
Let f : Ω → R be a continuously differentiable function.
Necessary and sufficient conditions for a local minimum:
x
∗
is a local minimum of f (x) if and only if
1 f has zero gradient at x
∗
:
∇
x
f(x
∗
) = 0
2 and the Hessian of f at w
∗
is positive semi-definite:
v
t
(∇
2
f(x
∗
)) v ≥ 0, ∀v ∈ R
n
where
∇
2
f(x) =
∂
2
f(x)
∂x
2
1
· · ·
∂
2
f(x)
∂x
1
∂x
n
.
.
.
.
.
.
.
.
.
∂
2
f(x)
∂x
n
∂x
1
· · ·
∂
2
f(x)
∂x
2
n
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