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Global+Optimization+Algorithms+Theory+and+Application_Part2

# Global+Optimization+Algorithms+Theory+and+Application_Part2...

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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

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