504 CHAPTER 23. OPTIMIZATION

(In this formula, the symbol ∑0k=1 ak will denote the number 0.)

Definition 23.7.2 The matrix(

∂ 2 f∂xi∂x j

(x))

is called the Hessian matrix, denoted

by H (x).

Now recall the Taylor formula with the Lagrange form of the remainder.

Theorem 23.7.3 Let h : (−δ ,1+δ )→R have m+1 derivatives. Then there existst ∈ (0,1) such that

h(1) = h(0)+m

∑k=1

h(k) (0)k!

+h(m+1) (t)(m+1)!

.

Now let f : U → R where U is an open subset of Rp. Suppose f ∈C2 (U). Let x ∈Uand let r > 0 be such that

B(x,r)⊆U.

Then for ||v||< r considerf (x+tv)− f (x)≡ h(t)

for t ∈ [0,1]. Then from Taylor’s theorem for the case where m = 2 and the chain rule,

h′ (t) = ∑i

∂ f∂xi

(x+ tv)vi,h′′ (t) = ∑j∑

i

∂ 2 f∂x j∂xi

(x+ tv)viv j.

Thush′′ (t) = vT H (x+ tv)v.

From Theorem 23.7.3 there exists t ∈ (0,1) such that

f (x+v) = f (x)+∂ f∂xi

(x)vi +12vT H (x+ tv)v

By the continuity of the second partial derivative

f (x+v) = f (x)+∇ f (x) ·v+12vT H (x)v+

12(vT (H (x+tv)−H (x))v

)(23.4)

where the last term satisfies

lim|v|→0

12

(vT (H (x+tv)−H (x))v

)|v|2

= 0 (23.5)

because of the continuity of the entries of H (x).

Theorem 23.7.4 Suppose x is a critical point for f . That is, suppose ∂ f∂xi

(x) = 0for each i. Then if H (x) has all positive eigenvalues, x is a local minimum. If H (x) has allnegative eigenvalues, then x is a local maximum. If H (x) has a positive eigenvalue, thenthere exists a direction in which f has a local minimum at x, while if H (x) has a negativeeigenvalue, there exists a direction in which f has a local maximum at x.