Page 281 - MATLAB an introduction with applications

266 ——— MATLAB: An Introduction with Applications

The conjugate directions in two dimensions are shown in Fig. 5.2. The first search direction is the
gradient vector, which is chosen arbitrarily. The second direction is conjugate to the first one. Several
methods have been proposed for generating the first search direction and these methods are applied to
non-quadratic functions also. It should be noted that Newton’s method would go to the minimum in one
step if the function to be minimized is quadratic.
x 2
1 x
Conjugate vector
C 0

0 x

Gradient
vectors

2 x

1 x
Fig. 5.2 Conjugate directions

5.5 POWELL’S METHOD

For an optimization problem involving n design variables, the basic algorithm is described below:
(a) select a point x in the design space
0
(b) select the starting vectors V , i = 1, 2, …, n
i
(c) do with i = 1, 2, …, n
minimize F(x) along the line thro x in the direction of V . i
i–1
assume the minimum point as x i
end do
(d) V n +1 ← x – x n
0
minimize F(x) along the line thro x in the direction of V n + 1
0
assume the minimum point as x n + 1
if |x n + 1 – x | < ∈ exit loop
0
do with i = 1, 2, …, n
V ← V i + 1
i
end do
end cycle
The minimum point of a quadratic surface is reached in n cycles.