Adaptive Scalarization Methods in Multiobjective Optimization

ISBN-13: 9783642098048 / Angielski / Miękka / 2010 / 241 str.

Gabriele Eichfelder

Opis

In many areas in engineering, economics and science new developments are only possible by the application of modern optimization methods. Theoptimizationproblemsarisingnowadaysinapplicationsaremostly multiobjective, i.e. many competing objectives are aspired all at once. These optimization problems with a vector-valued objective function have in opposition to scalar-valued problems generally not only one minimal solution but the solution set is very large. Thus the devel- ment of e?cient numerical methods for special classes of multiobj- tive optimization problems is, due to the complexity of the solution set, of special interest. This relevance is pointed out in many recent publications in application areas such as medicine ( 63, 118, 100, 143]), engineering( 112,126,133,211,224], referencesin 81]), environmental decision making ( 137, 227]) or economics ( 57, 65, 217, 234]). Consideringmultiobjectiveoptimizationproblemsdemands?rstthe de?nition of minimality for such problems. A ?rst minimality notion traces back to Edgeworth 59], 1881, and Pareto 180], 1896, using the naturalorderingintheimagespace.A?rstmathematicalconsideration ofthistopicwasdonebyKuhnandTucker 144]in1951.Sincethattime multiobjective optimization became an active research ?eld. Several books and survey papers have been published giving introductions to this topic, for instance 28, 60, 66, 76, 112, 124, 165, 188, 189, 190, 215]. Inthelastdecadesthemainfocuswasonthedevelopmentofinteractive methods for determining one single solution in an iterative process.

Recenzje

From the reviews:

"This monograph on vector optimization is besides theoretical basics ... dedicated to skillful approximations of the whole efficient set. ... This monograph shows experience of the author in the big field of the contents of this book and can be highly recommended to all who work (or want to work) with multicriteria problems." (Alfred Göpfert, Zentralblatt MATH, Vol. 1145, 2008)

Spis treści

Theory.- Theoretical Basics of Multiobjective Optimization.- Scalarization Approaches.- Sensitivity Results for the Scalarizations.- Numerical Methods and Results.- Adaptive Parameter Control.- Numerical Results.- Application to Intensity Modulated Radiotherapy.- Multiobjective Bilevel Optimization.- Application to Multiobjective Bilevel Optimization.

Blurb

This book presents new adaptive solution methods for multiobjective optimization problems based on parameter dependent scalarizations. With the help of sensitivity results an adaptive parameter control is developed so that high-quality approximations of the efficient set are generated. These examinations are based on a general scalarization approach for arbitrary partial orderings defined by a closed pointed convex cone in the objective space. The application of the results to many other well-known scalarization
methods is also presented. Background material of multiobjective optimization and scalarization approaches is concisely summarized at the beginning. The effectiveness of these new methods is demonstrated by test problems and a recent problem in intensity-modulated radiotherapy. The book concludes with a further application: a procedure for solving multiobjective bilevel optimization problems.