5.2 Unimodal Function 253
Table 5.1 One-dimensional Minimization Methods
Elimination
methods
Unrestricted
search Requiring no
derivatives
(quadratic)
Requiring
derivatives
Cubic
Direct root
Newton
Quasi-Newton
Secant
Exhaustive search
Dichotomous
search
Fibonacci method
Golden section
method
Analytical methods Numerical methods
(differential calculus methods)
Interpolation
methods
interpolation methods involve polynomial approximations to the given function. The
direct root methods are root finding methods that can be considered to be equivalent
to quadratic interpolation.
5.2 Unimodal Function
Aunimodal functionis one that has only one peak (maximum) or valley (minimum)
in a given interval. Thus a function of one variable is said to beunimodalif, given
that two values of the variable are on the same side of the optimum, the one nearer
the optimum gives the better functional value (i.e., the smaller value in the case of a
minimization problem). This can be stated mathematically as follows:
A functionf (x) is unimodal if(i) x 1 < x 2 < x∗implies thatf(x 2 ) <
f(x 1 ) and (ii), x 2 >x 1 >x∗implies thatf(x 1 ) < f (x 2 ) where, x∗is the
minimum point.
Some examples of unimodal functions are shown in Fig. 5.4. Thus a unimodal function
can be a nondifferentiable or even a discontinuous function. If a function is known to
be unimodal in a given range, the interval in which the minimum lies can be narrowed
down provided that the function values are known at two different points in the range.
Figure 5.4 Unimodal function.
