254 Nonlinear Programming I: One-Dimensional Minimization Methods
Figure 5.5 Outcome of first two experiments: (a) f 1 < f 2 ; (b) f 1 >f 2 ; (c) f 1 =f 2.For example, consider the normalized interval [0, 1] and two function evaluations
within the interval as shown in Fig. 5.5. There are three possible outcomes, namely,
f 1 < f 2 ,f 1 >f 2 , orf 1 =f 2. If the outcome is thatf 1 < f 2 , the minimizingxcannot
lie to the right ofx 2. Thus that part of the interval [x 2 , 1] can be discarded and a new
smaller interval of uncertainty, [0,x 2 ], results as shown in Fig. 5.5a.Iff (x 1 ) >f(x 2 ),
theinterval [0,x 1 ] can be discarded to obtain a new smaller interval of uncertai nty,
[x 1 , 1] (Fig. 5.5b), while iff (x 1 ) =f(x 2 ) intervals [0,, x 1 ] and [x 2 , 1] can both be
discarded to obtain the new interval of uncertainty as [x 1 , x 2 ] (Fig. 5.5c). Further,
if one of the original experiments†remains within the new interval, as will be the
situation in Fig. 5.5aandb, only one other experiment need be placed within the new
interval in order that the process be repeated. In situations such as Fig. 5.5c, two more
experiments are to be placed in the new interval in order to find a reduced interval of
uncertainty.
The assumption of unimodality is made in all the elimination techniques. If a
function is known to bemultimodal(i.e., having several valleys or peaks), the range of
the function can be subdivided into several parts and the function treated as a unimodal
function in each part.ELIMINATION METHODS
5.3 Unrestricted Search
In most practical problems, the optimum solution is known to lie within restricted
ranges of the design variables. In some cases this range is not known, and hence the
search has to be made with no restrictions on the values of the variables.5.3.1 Search with Fixed Step Size
The most elementary approach for such a problem is to use a fixed step size and move
from an initial guess point in a favorable direction (positive or negative). The step size†Each function evaluation is termed as an experiment or a trialin the elimination methods.