As optimization methods become widely used in engineering, economics and other sciences, an increasing number of problems are encountered that cannot be solved successfully by standard techniques of linear and nonlinear programming. These are nonconvex global optimization problems whose distinguishing feature is multiextremality, i.e. the presence of many local optima which fail to be global. In this Chapter we shall introduce various models of global optimization arising from applications and show the common general mathematical structure underlying all these problems, despite their great diversity. This general mathematical structure gives insight into the nature of global optimization and provides the foundation for a unified approach which consists in reducing every nonconvex global optimization problem to a canonical form, namely: minimizing (or maximizing) a linear function over a difference of two convex sets. While many general purpose algorithms of global optimization are available (cf. Horst and Tuy (1993), most of them are able to solve practically only problem instances of very limited size, as would be expected from the NP-hardness of these problems. On the other hand, the physical context of the model usually imposes additional structures on the problem, making it much more tractable. Therefore, for the applications it is important to study problems with special structures, yet of relevant practical interest. Most of these problems can be decomposed into subproblems whose nonconvex component is located in a space of low dimension and hence can be efficiently solved by specialized algorithms.
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- Scope of Global Optimization
Phan Thien Thach
- Springer US
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