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CHAPTER 5 5
5

Optimization

5.1 INTRODUCTION

Optimization is minimizing or maximizing a function. The function F(x) is called the merit function or objective
function. The components of x are known as the design variables.
The minimum point must be bracketed before a minimization algorithm can be used. The bracketing
procedure consists of starting with an initial value of x and moving downhill computing the functions at
0
x , x , x , … until the point x is reached where f (x) increases for the first time. The minimum point is now
3
2
n
1
bracketed in the interval (x n–2 , x ). The increasing in step size follows as h i + 1 = c h where c > 1.
i
n
Suppose the minimum of f(x) has been bracketed in the interval (a, b) of length h. To telescope the
interval, the function at x = b – Rh and x = a + Rh is evaluated. If f > f , then the minimum lies in (x , b);
2
2
1
1
1
otherwise it is located in (a, x ).
2
Next, we evaluate the function at x = a + Rh′ and repeat the process. We noted that x – x = 2 Rh – h
2
1
2
and x – a = h′ – Rh′ or 2Rh – h = h′ – Rh′ and substituting h′ = Rh, we obtain R = 0.618033989. The number
1
of telescopings required to reduce h from |b – a| to an error tolerance ∈ is given by
ln ( / | ba∈ − ) | ∈
n = = − 2.078087 ln .
−
lnR | b a |
5.2 CONJUGATE GRADIENT METHODS
The objective here is to minimize F(x), where the components of x are the n independent design variables.
Consider the quadratic function
() =−
Fx c ∑ b x + 1 ∑∑ A x x j
ii
ij i
i 2 i j
1
=− T x Ax ...(5.1)
cb x +
T
2