Adaptive Scalarization Methods in Multiobjective Optimization (eBook)

Preface 7
Contents 11
Part I Theory 15
1 Theoretical Basics of Multiobjective Optimization 16
1.1 Basic Concepts 16
1.2 Polyhedral Ordering Cones 28
2 Scalarization Approaches 34
2.1 Pascoletti-Serafini Scalarization 36
2.2 Properties of the Pascoletti-Serafini Scalarization 38
2.3 Parameter Set Restriction for the Pascoletti-Serafini Scalarization 44
2.4 Modified Pascoletti-Serafini Scalarization 57
2.5 Relations Between Scalarizations 62
3 Sensitivity Results for the Scalarizations 80
3.1 Sensitivity Results in Partially Ordered Spaces 81
3.2 Sensitivity Results in Naturally Ordered Spaces 96
3.3 Sensitivity Results for the 107
Part II Numerical Methods and Results 112
4 Adaptive Parameter Control 114
4.1 Quality Criteria for Approximations 114
4.2 Adaptive Parameter Control in the Bicriteria Case 120
4.3 Adaptive Parameter Control in the Multicriteria Case 147
5 Numerical Results 154
5.1 Bicriteria Test Problems 154
5.2 Tricriteria Test Problems 168
6 Application to Intensity Modulated Radiotherapy 180
6.1 Problem Formulation Using a Bicriteria Approach 181
6.2 Problem Formulation Using a Tricriteria Approach 189
Part III Multiobjective Bilevel Optimization 194
7 Application to Multiobjective Bilevel Optimization 196
7.1 Basic Concepts of Bilevel Optimization 197
7.2 Induced Set Approximation 199
7.3 Induced Set Refinement 206
7.4 Algorithm 210
7.5 Numerical Results 212
7.6 Multiobjective Bilevel Optimization Problems with Coupled Upper Level Constraints 223
References 232
Index 252