1
Introduction
One of the most fundamental principles in our world is the search for an optimal state.
It begins in the microcosm where atoms in physics try to form bonds
1
in order to minimize
the energy of their electrons [1625]. When molecules form solid bodies during the process of
freezing, they try to assume energy-optimal crystal structures. These processes, of course,
are not driven by any higher intention but purely result from the laws of physics.
The same goes for the biological principle of
survival of the fittest
[1940] which, together
with the biological evolution [485], leads to better adaptation of the species to their environ-
ment. Here, a local optimum is a well-adapted species that dominates all other animals in
its surroundings. Homo sapiens have reached this level, sharing it with ants, bacteria, ﬂies,
cockroaches, and all sorts of other creepy creatures.
As long as humankind exists, we strive for perfection in many areas. We want to reach
a maximum degree of happiness with the least amount of effort. In our economy, profit and
sales must be maximized and costs should be as low as possible. Therefore, optimization is
one of the oldest of sciences which even extends into daily life [1519].
If something is important, general, and abstract enough, there is always a mathematical
discipline dealing with it. Global optimization
2
is the branch of applied mathematics and nu-
merical analysis that focuses on, well, optimization. The goal of global optimization is to find
the best possible elements
x
⋆
from a set
X
according to a set of criteria
F
=
{
f
1
,f
2
,..,f
n
}
.
These criteria are expressed as mathematical functions
3
, the so-called objective functions.
Definition 1.1 (Objective Function).
An objective function
f
:
X
mapsto→
Y
with
Y
⊆
R
is
a mathematical function which is subject to optimization.
The codomain
Y
of an objective function as well as its range must be a subset of the real
numbers (
Y
⊆
R
). The domain
X
of
f
is called problem space and can represent any type
of elements like numbers, lists, construction plans, and so on. It is chosen according to the
problem to be solved with the optimization process. Objective functions are not necessarily
mere mathematical expressions, but can be complex algorithms that, for example, involve
multiple simulations. Global optimization comprises all techniques that can be used to find
the best elements
x
⋆
in
X
with respect to such criteria
f
∈
F
.
In the remaining text of this introduction, we will first provide a rough classification
of the different optimization techniques which we will investigate in the further course of
this book (
Section 1.1
). In
Section 1.2
, we will outline how these
best
elements which we
are after can be defined. We will use
Section 1.3
to shed some more light onto the meaning
and inter-relation of the symbols already mentioned (
f
,
F
,
x
,
x
⋆
,
X
,
Y
, . . . ) and outline
1
http://en.wikipedia.org/wiki/Chemical_bond
[accessed 2007-07-12]
2
http://en.wikipedia.org/wiki/Global_optimization
[accessed 2007-07-03]
3