Adaptive Scalarization Methods in Multiobjective Optimization

By: Gabriele Eichfelder

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Hardcover | 4 June 2008

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Hardcover
256 Pages

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23.39 x 15.6 x 1.6

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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.

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"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 G¶pfert, Zentralblatt MATH, Vol. 1145, 2008)

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You Can Find This Book In

Non-Fiction Business & Management Operational Research Computing & I.T.Databases Data Capture & Analysis Computer Science Mathematical Theory of Computation Mathematics Probability & Statistics Optimisation Linear Programming

Theory
Theoretical Basics of Multiobjective Optimization	p. 3
Basic Concepts	p. 3
Polyhedral Ordering Cones	p. 15
Scalarization Approaches	p. 21
Pascoletti-Serafini Scalarization	p. 23
Properties of the Pascoletti-Serafini Scalarization	p. 25
Parameter Set Restriction for the Pascoletti-Serafini Scalarization	p. 31
Bicriteria Case	p. 32
General Case	p. 40
Modified Pascoletti-Serafini Scalarization	p. 44
Relations Between Scalarizations	p. 49
¿-Constraint Problem	p. 49
Normal Boundary Intersection Problem	p. 53
Modified Polak Problem	p. 55
Weighted Chebyshev Norm Problem	p. 57
Problem According to Gourion and Luc	p. 58
Generalized Weighted Sum Problem	p. 59
Weighted Sum Problem	p. 61
Problem According to Kaliszewski	p. 65
Further Scalarizations	p. 66
Sensitivity Results for the Scalarizations	p. 67
Sensitivity Results in Partially Ordered Spaces	p. 68
Sensitivity Results in Naturally Ordered Spaces	p. 83
Sensitivity Results for the ¿-Constraint Problem	p. 94
Numerical Methods and Results
Adaptive Parameter Control	p. 101
Quality Criteria for Approximations	p. 101
Adaptive Parameter Control in the Bicriteria Case	p. 107
Algorithm for the Pascoletti-Serafini Scalarization	p. 123
Algorithm for the ¿-Constraint Scalarization	p. 128
Algorithm for the Normal Boundary Intersection Scalarization	p. 130
Algorithm for the Modified Polak Scalarization	p. 132
Adaptive Parameter Control in the Multicriteria Case	p. 134
Numerical Results	p. 141
Bicriteria Test Problems	p. 141
Test Problem 1: ¿-Constraint Scalarization	p. 141
Test Problem 2: Comparison with the Weighted Sum Method	p. 144
Test Problem 3: Non-Convex Image Set	p. 147
Test Problem 4: Non-Connected Efficient Set	p. 149
Test Problem 5: Various Ordering Cones	p. 153
Tricriteria Test Problems	p. 155
Test Problem 6: Convex Image Set	p. 155
Test Problem 7: Non-Convex Image Set	p. 158
Test Problem 8: Comet Problem	p. 161
Test Problem 9: Non-Connected Efficient Set	p. 163
Application to Intensity Modulated Radiotherapy	p. 167
Problem Formulation Using a Bicriteria Approach	p. 168
Problem Formulation Using a Tricriteria Approach	p. 176
Multiobjective Bilevel Optimization
Application to Multiobjective Bilevel Optimization	p. 183
Basic Concepts of Bilevel Optimization	p. 184
Induced Set Approximation	p. 186
Induced Set Refinement	p. 193
Algorithm	p. 197
Numerical Results	p. 199
Test Problem	p. 199
Application Problem	p. 202
Multiobjective Bilevel Optimization Problems with Coupled Upper Level Constraints	p. 210
References	p. 219
Index	p. 239
Table of Contents provided by Publisher. All Rights Reserved.

Adaptive Scalarization Methods in Multiobjective Optimization

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