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Über dieses Buch

Global optimization is concerned with finding the global extremum (maximum or minimum) of a mathematically defined function (the objective function) in some region of interest. In many practical problems it is not known whether the objective function is unimodal in this region; in many cases it has proved to be multimodal. Unsophisticated use of local optimization techniques is normally inefficient for solving such problems. Therefore, more sophisticated methods designed for global optimization, i.e. global optimization methods, are important from a practical point of view. Most methods discussed here assume that the extremum is attained in the interior of the region of interest, i.e., that the problem is essentially unconstrained. Some methods address the general constrained problem. What is excluded is the treatment of methods designed for problems with a special structure, such as quadratic programming with negatively quadratic forms. This book is the first broad treatment of global optimization with an extensive bibliography covering research done both in east and west. Different ideas and methods proposed for global optimization are classified, described and discussed. The efficiency of algorithms is compared by using both artificial test problems and some practical problems. The solutions of two practical design problems are demonstrated and several other applications are referenced. The book aims at aiding in the education, at stimulating the research in the field, and at advising practitioners in using global optimization methods for solving practical problems.



1. Introduction

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2. Covering methods

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3. Methods of generalized descent

The idea behind these methods is rather simple and obvious. Their algorithmical realization may be reduced to the choice a of local minimization algorithm and of auxiliary functions whose minimization gives conditions under which better local minima of f(·) may successively be found. The auxilary functions are constructed using the objective function and penalty functions.
However, the auxilary functions normally become rather flat and are therefore difficult to minimize. An exception to this are algebraic objective functions but in practice such functions are rare.
Further research is necessary to define more precisely the class of functions to which the algorithms of the considered type may be efficiently applied. An important problem to be solved is how to ensure the numerical stability of the exploited local minimization algorithms when applied to flat functions.

4. Random search methods

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5. Clustering methods

Clustering methods have proved very successful in tackling the global optimization problem. One reason is that they make it possible to very efficiently combine global and local search.
The clustering technique facilitate efficient representation of the information on the problem encountered in the solution process. The output from the process showing the evolution of clusters contains a lot of extra information hard to formalize, but of great importance for the practitioner solving a real world problem.
In clustering methods stopping conditions available for uniform sampling and multistart relating the goal prescribed to the effort needed to achieve this goal can be used. Considerable theoretical progress has been made in this direction. The clustering approach to global optimization thus rests on a sound theoretical base which makes these methods attractive both from a practical and mathematical standpoint.

6. Methods based on statistical models of objective functions

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7. Miscellaneous

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8. Testing and applications

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