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Global optimization in action. Continuous and Lipschitz optimization: algorithms, implementations and applications. (English) Zbl 0842.90110

Nonconvex Optimization and Its Applications. 6. Dordrecht: Kluwer Academic Publishers. xxvii, 478 p. (1996).
The present volume is an interesting textbook on the treatment of a broad class of global optimization problems. In four chapters the author gives a detailed outline of the theoretical foundations of such problems, about several optimization algorithms – including special implementation aspects – and about many examples regarding the application of the described methods to many practical decision making problems.
In detail, chapter 1 (38p.) contains a short introduction to the subject and some theoretical remarks. So the most important specifications of global optimization – e.g. concave minimization, reverse convex programming, d.c. programming, multiplicative programming, fractional programming, combinatorial optimization, Lipschitz global optimization – are discussed and compared. Dependent on the special structure of these problems the author presents a brief overview on known global solution procedures: uniform grid search, random search, tunnelling techniques, path-following methods, homotopy methods, relaxation strategies (closely connected with cutting plane methods) and branch-and-bound methods. In this place the global character of the solution concept is emphasized. It is carried out that this concept is (much) more difficult than its local counterpart which can be handled by means of classical analytical tools. Most of the global search strategies are based on adaptive, sequential partition and search in the whole feasible set to find approximate solutions. Frequently, the procedures are combined with local methods to refine the result.
In Chapter 2 (106 pages, the main part of the book) the author gives a detailed description of partition strategies in global optimization. All the presented algorithms can be summarized shortly in the following search scheme: Initialization \(\to\) Sampling \(\to\) Selection \(\to\) Partition refinement \(\to\) Stopping criterion.
The efficiency of the algorithm depends essentially on the kind of selection which is determined by a special partition operator. For this the author gives a general approach. So-called regular partition strategies are discussed for which necessary and sufficient conditions but also efficiency estimations can be formulated. By means of special algorithms – e.g. the Piyavskij algorithms and the Kushner algorithm – the general assertions are illustrated. The chapter is structured in such a manner that starting with the one-dimensional case the feasible set will be generalized step by step (multi-dimensional intervals, simplices, convex sets, star-shaped sets, level sets of Lipschitz functions.
Chapter 3 (92 p.) contains some further theoretical results and important aspects of efficient implementations of the described partition strategies. So, e.g. questions about the estimation of the Lipschitz constants and the inclusion of penalty techniques in the search strategies are discussed. Regarding the treatment of stochastic optimization problems (decision making with uncertainty) the author presents special stochastic procedures based on random search strategies. A special program system for the solution of general Lipschitz optimization problems is introduced in a separate subsection.
In the last chapter (with 186 pages the most extensive chapter) the author discusses many existing and prospective applications of the described approach to practical decision making models. The problem classes originate from several areas of general interest in numerical mathematics and – in a large scale – from the increasingly important field of environmental systems modelling and management: theory of nonlinear equalities and inequalities, approximation theory, data classification, facility location, aggregation of expert opinions, product design and calibration of complex system models with important practical specifications (discussion of wastwater model, sediment-water model, ground water model, river flow model, chemical fate model, aquifer model, lake eutrophication model, …). In the end of the chapter the author gives a short review of numerous other fields in which global optimization can be applied.
The book is written mainly for readers working in the field of operations research, management sciences and engineering, for researcher, teacher and students which are active in mathematical optimization and – last not least – for researchers and other professionals who apply global optimization in their own research field. A large bibliographical list at the end of the book refers to further literature of the subject.

MSC:

90C30 Nonlinear programming
90-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming
49M37 Numerical methods based on nonlinear programming
90C90 Applications of mathematical programming
49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
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