principles and applications of microearthquake networks

5.4. Nonlinear Optimization 123
Fletcher, 1980; Gillet al., 1981). Unfortunately, no single method has been
found to solve all problems effectively. The subject is being actively pursued
by scientists in many disciplines. In this section, we briefly describe
some elementary aspects of nonlinear optimization.
5.4.1. Problem DeJinition
The basic mathematical problem in optimization is to minimize a scalar
quantity $which is the value of a function F(xl, x2, . . . , xn) of n independent
variables. These independent variables, xl, x2, . . . , xn, must be
adjusted to obtain the minimum required, ie.,
(5.64) minimize {$ = F(x,, x2, . . . , x,)}
The function Fis referred to as the objective function because its value +
is the quantity to be minimized.
It is useful to consider the n independent variables, xl, x2, . . . , xn, as a
vector in n-dimensional Euclidean space,
(5.65)
where the superscript T denotes the transpose of a vector or a matrix.
During the optimization process, the coordinates of x will take on successive
values as adjustments are made. Each set of adjustments to x is
termed an iteration, and a number of iterations are generally required
before a minimum is reached. In order to start an iterative procedure, an
initial estimate of x must be given. After K iterations, we denote the value
of $ by $Ltm, and the value of x by x(~). Changes in the value of x between
two successive iterations are just the adjustments applied. These adjustments
may also be thought of as components of an adjustment vector,
(5.66) 6x = (6x1, 652, . . * , 6Xn)T
The goal of optimization is to find after K iterations a xCK) that gives a
minimum value $(K) of the objective function F. A point x(~) is called a
global minimum if it gives the lowest possible value of F. In general, a
global minimum need not be unique; and in practice, it is very difficult to
tell if a global minimum has been reached by an iterative procedure. We
may only claim that a minimum within a local area of search has been
obtained. Even such a local minimum may not be unique locally.
Methods in optimization may be divided into three classes: (1) search
methods which use function evaluation only; (2) methods which in addition
require gradient information or first derivatives; and (3) methods
which require function, gradient, and second-derivative information. The
appeal of each class depends on the particular problem and the available