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This monograph deals with a general class of solution approaches in deterministic global optimization, namely the geometric branch-and-bound methods which are popular algorithms, for instance, in Lipschitzian optimization, d.c. programming, and interval analysis.It also introduces a new concept for the rate of convergence and analyzes several bounding operations reported in the literature, from the theoretical as well as from the empirical point of view. Furthermore, extensions of the prototype algorithm for multicriteria global optimization problems as well as mixed combinatorial optimization problems are considered. Numerical examples based on facility location problems support the theory. Applications of geometric branch-and-bound methods, namely the circle detection problem in image processing, the integrated scheduling and location makespan problem, and the median line location problem in the three-dimensional space are also presented.
The book is intended for both researchers and students in the areas of mathematics, operations research, engineering, and computer science.
List of contents
Principals and basic concepts.- The geometric branch-and-bound algorithm.-Bounding operations.- Extension for multicriteria problems.- Multicriteria discarding tests.- Extension for mixed combinatorial problems.- The circle detection problem.- Integrated scheduling and location problems.- The median line problem.- Summary and discussion.- References.- Index.
Summary
This monograph deals with a general class of solution approaches in deterministic global optimization, namely the geometric branch-and-bound methods which are popular algorithms, for instance, in Lipschitzian optimization, d.c. programming, and interval analysis.It also introduces a new concept for the rate of convergence and analyzes several bounding operations reported in the literature, from the theoretical as well as from the empirical point of view. Furthermore, extensions of the prototype algorithm for multicriteria global optimization problems as well as mixed combinatorial optimization problems are considered. Numerical examples based on facility location problems support the theory. Applications of geometric branch-and-bound methods, namely the circle detection problem in image processing, the integrated scheduling and location makespan problem, and the median line location problem in the three-dimensional space are also presented.
The book is intended for both researchers and students in the areas of mathematics, operations research, engineering, and computer science.
Additional text
From the reviews:
“This little book gives a unified treatment and current state of affairs of geometric branch-and-bound methods for global optimization in low dimensions and their applications. … a valuable tool for anyone interested in the solution of low-dimensional global optimization problems. The whole text is very carefully written with special attention given to the introduction of economical notation and corresponding definitions that considerably shorten the arguments. This is particularly useful for students, but also for the working specialist.” (Frank Plastria, Mathematical Reviews, September, 2013)
“The book presents a collection of papers by Scholz and other authors, supplemented with some auxiliary material and discussion, and includes new details, newly edited text, and revised proofs. … an expert in global optimization and an intended user of global optimization software would benefit from owning this book on geometric B&B methods, which a recognized expert has researched.” (Antanas Zilinskas, Interfaces, Vol. 43 (1), 2013)
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From the reviews:
"This little book gives a unified treatment and current state of affairs of geometric branch-and-bound methods for global optimization in low dimensions and their applications. ... a valuable tool for anyone interested in the solution of low-dimensional global optimization problems. The whole text is very carefully written with special attention given to the introduction of economical notation and corresponding definitions that considerably shorten the arguments. This is particularly useful for students, but also for the working specialist." (Frank Plastria, Mathematical Reviews, September, 2013)
"The book presents a collection of papers by Scholz and other authors, supplemented with some auxiliary material and discussion, and includes new details, newly edited text, and revised proofs. ... an expert in global optimization and an intended user of global optimization software would benefit from owning this book on geometric B&B methods, which a recognized expert has researched." (Antanas Zilinskas, Interfaces, Vol. 43 (1), 2013)
