Engineering Optimization: Theory and Practice, Fourth Edition

258 Nonlinear Programming I: One-Dimensional Minimization Methods

Figure 5.7 Dichotomous search.

function at the two points, almost half of the interval of uncertainty is eliminated. Let

the positions of the two experiments be given by (Fig. 5.7)

x 1 =

L 0

2

−

δ

2

x 2 =

L 0

2

+

δ

2

whereδis a small positive number chosen so that the two experiments give significantly

different results. Then the new interval of uncertainty is given by (L 0 /2 +δ/2). The

building block of dichotomous search consists of conducting a pair of experiments

at the center of the current interval of uncertainty. The next pair of experiments is,

therefore, conducted at the center of the remaining interval of uncertainty. This results

in the reduction of the interval of uncertainty by nearly a factor of 2. The intervals

of uncertainty at the end of different pairs of experiments are given in the following

table:

Number of experiments 2 4 6

Final interval of uncertainty

1

2

(L 0 +δ)

1

2

(

L 0 +δ

2

)

+

δ

2

1

2

(

L 0 +δ

4

+

δ

2

)

+

δ

2

In general, the final interval of uncertainty after conductingnexperiments (neven) is

given by

Ln=

L 0

2 n/^2

+δ

(

1 −

1

2 n/^2

)

(5.3)

The following example is given to illustrate the method of search.

Example 5.5 Find the minimum off=x(x− 1 .5) in the interval (0.0, 1.00) to within

10% of the exact value.

SOLUTION The ratio of final to initial intervals of uncertainty is given by [from

Eq. (5.3)]

Ln

L 0

=

1

2 n/^2

+

δ

L 0

(

1 −

1

2 n/^2

)