Page 282 - MATLAB an introduction with applications

Optimization ——— 267

5.6 FLETCHER-REEVES METHOD

The algorithm is described below:
(a) select a starting point x 0
(b) g ← – ∇F(x )
0
0
(c) V ← g 0
0
(d) loop with i = 0, 1, 2, …
minimize F(x) along V ; x i + 1 is the minimum point assumed
i
g i + 1 ← –∇F(x i + 1 )
if |g i + 1 | < ∈ or |F(x i + 1 ) – F(x )| < ∈ exit loop
i
r ← (g i + 1 . g i + 1 )/(g . g )
i
i
V i + 1 ← g i + 1 + rV i
end loop
The method will find the minimum of a quadratic function in n iterations. Also, V and V i + 1 are mutually
i
conjugate. In other words,
T
VAV i+ 1 = 0 and gg i i+ 1 = 0 .
i
i
5.7 HOOKE AND JEEVES METHOD

Hooke and Jeeves method is a sequential technique each step of which consists of two kinds of moves,
the exploratory move and the pattern move.
General procedure:

x

1

x

2
1. Start with an arbitrarily chosen point X =  , called the starting base point and prescribed step
1



x

n
lengths ∆x in each of the coordinate directions u , i = 1, 2, …, n. Set k = 1.
i
i
2. Compute f = f (X ). Set i = 1, Y = X , and start the exploratory move as stated in step 3.
k
k
k
k0
3. The variable x is perturbed about the current temporary base point Y k, i–1 to obtain the new temporary
i
base point as
 Y +∆ x u if f + = ( f Y +∆ x u )
,1
,1
 ki− i i ki− i i
 < f = ( f Y ki ,1 − )
 −
 Y ki− −∆ x u if f = ( f Y ki− −∆ x u )
i i
,1
i i
,1
Y , ki = 
 < f = ( f Y ki ,1 − )
 +
 < f = ( f Y ki− + ∆ x u )
,1
i i
 Y if f = ( f Y ) < min( f + , f − )
 ki− ki−
,1
,1