5.3 Unrestricted Search 255
used must be small in relation to the final accuracy desired. Although this method is
very simple to implement, it is not efficient in many cases. This method is described
in the following steps:
1.Start with an initial guess point, say,x 1.
2 .Findf 1 = f(x 1 ).
3 .Assuming a step sizes, findx 2 =x 1 +s.
4 .Findf 2 = f(x 2 ).
5 .Iff 2 < f 1 , and if the problem is one of minimization, the assumption of uni-
modality indicates that the desired minimum cannot lie atx < x 1. Hence the
search can be continued further along pointsx 3 , x 4 ,... using the unimodality
assumption while testing each pair of experiments. This procedure is con-
tinued until a point, xi=x 1 + (i− 1 )s, shows an increase in the function
value.
6.The search is terminated atxi, and eitherxi− 1 orxican be taken as the optimum
point.
7.Originally, iff 2 >f 1 , the search should be carried in the reverse direction at
pointsx− 2 , x− 3 ,... , wherex−j=x 1 − (j− 1 )s.
8.Iff 2 =f 1 , the desired minimum lies in betweenx 1 andx 2 , and the minimum
point can be taken as eitherx 1 orx 2.
9 .If it happens that bothf 2 andf− 2 are greater thanf 1 , it implies that the desired
minimum will lie in the double intervalx− 2 < x < x 2.
5.3.2 Search with Accelerated Step Size
Although the search with a fixed step size appears to be very simple, its major limitation
comes because of the unrestricted nature of the region in which the minimum can lie.
For example, if the minimum point for a particular function happens to bexopt=
50 ,000 and, in the absence of knowledge about the location of the minimum, ifx 1 and
sare chosen as 0.0 and 0.1, respectively, we have to evaluate the function 5,000,001
times to find the minimum point. This involves a large amount of computational work.
An obvious improvement can be achieved by increasing the step size gradually until
the minimum point is bracketed. A simple method consists of doubling the step size
as long as the move results in an improvement of the objective function. Several other
improvements of this method can be developed. One possibility is to reduce the step
length after bracketing the optimum in (xi− 1 , xi) By starting either from. xi− 1 orxi,
the basic procedure can be applied with a reduced step size. This procedure can be
repeated until the bracketed interval becomes sufficiently small. The following example
illustrates the search method with accelerated step size.
Example 5.3 Find the minimum off=x(x− 1 .5) by starting from 0.0 with an initial
step size of 0.05.
SOLUTION The function value atx 1 isf 1 = 0. 0. If we try to start moving in the
negativexdirection, we find thatx− 2 = − 0. 0 5 andf− 2 = 0. 0 775. Sincef− 2 >f 1 , the
assumption of unimodality indicates that the minimum cannot lie toward the left of
x− 2. Thus we start moving in the positivexdirection and obtain the following results:
