Numerical Methods and Optimization

Unconstrained Optimization 327

Returning to the original problem (13.9)–(13.11), we have

ρ

N

=1/2=1−

F

1

F

2

;

ρ

N−1

=

1 − ρ

N

2 − ρ

N

=

F

1

F

3

=1−

F

2

F

3

;

.

.

.

ρ

k+1

=1−

F

N−k

F

N−k+1

;

ρ

k

=

1 − ρ

k+1

2 − ρ

k+1

=

F

N−k

F

N−k+2

=1−

F

N−k+1

F

N−k+2

;

.

.

.

ρ

1

=1−

F

N

F

N+1

.

Note that the above ρ

k

,k =1,...,N satisfy the conditions (13.10)–(13.11)

and thus represent the (unique) optimal solution to (13.9)–(13.11). 

13.3 Algorithmic Strategies for Unconstrained

Optimization

Consider the problem

minimizef(x).

Classical algorithms for this problem usually aim to construct a sequence of

points {x

(k)

: k ≥ 0}, such that x

(k)

→ x

∗

,k →∞,wherex

∗

is a stationary

point of f(x) (that is, ∇f (x

∗

) = 0). Each next point in this ...