Numerical Methods and Optimization

322 Numerical Methods and Optimization: An Introduction

If we additionally assume that ∇

2

f(x

∗

) is positive definite, then by Rayleigh’s

inequality,

1

2

d

T

∇

2

f(x

∗

)d ≥ λ

min

d

2

= λ

min

> 0.

Here λ

min

denotes the smallest eigenvalue of ∇

2

f(x

∗

). Thus, there exists >0

such that for any α ∈ (0,)wehavef(x

∗

+ αd) − f(x

∗

) > 0. Since d is an

arbitrary direction in IR

n

, x

∗

is a point of strict local minimum by definition.

Thus, the FONC, SONC, together with the positive definiteness of ∇

2

f(x

∗

),

constitute the sufficient conditions for a strict local minimizer. 

Example 13.3 Consider the function f (x)=x

3

1

−x

3

2

+3x

1

x

2

. We apply the

optimality conditions above to find its loc