A Book by Kaisa Miettinen:

Nonlinear Multiobjective Optimization by Kaisa Miettinen

Nonlinear Multiobjective Optimization

Kluwer Academic Publishers, Boston, 1999

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Description

Problems with multiple objectives and criteria are generally known as multiple criteria optimization or multiple criteria decision-making (MCDM) problems. So far, these types of problems have typically been modelled and solved by means of linear programming. However, many real-life phenomena are of a nonlinear nature, which is why we need tools for nonlinear programming capable of handling several conflicting or incommensurable objectives. In this case, methods of traditional single objective optimization and linear programming are not enough; we need new ways of thinking, new concepts, and new methods --- nonlinear multiobjective optimization.

Nonlinear Multiobjective Optimization provides an extensive, up-to-date, self-contained and consistent survey, review of the literature and of the state of the art on nonlinear (deterministic) multiobjective optimization, its methods, its theory and its background. The amount of literature on multiobjective optimization is immense. The treatment in this book is based on approximately 1500 publications in English printed mainly after the year 1980.

Problems related to real-life applications often contain irregularities and nonsmoothnesses. The treatment of nondifferentiable multiobjective optimization in the literature is rather rare. For this reason, this book contains material about the possibilities, background, theory and methods of nondifferentiable multiobjective optimization as well.

This book is intended for both researchers and students in the areas of (applied) mathematics, engineering, economics, operations research and management science; it is meant for both professionals and practitioners in many different fields of application. The intention has been to provide a consistent summary that may help in selecting an appropriate method for the problem to be solved. It is hoped the extensive bibliography will be of value to researchers.

Contents

PREFACE xiii
ACKNOWLEDGEMENTS xix
NOTATION AND SYMBOLS xxi
Part I TERMINOLOGY AND THEORY
1. INTRODUCTION 3
2. CONCEPTS
5
2.1 Problem Setting and General Notation
5
2.1.1. Multiobjective Optimization Problem
5
2.1.2. Background Concepts
6
2.2. Pareto Optimality
10
2.3. Decision Maker
14
2.4. Ranges of the Pareto Optimal Set
15
2.4.1. Ideal Objective Vector
15
2.4.2. Nadir Objective Vector
16
2.4.3. Related Topics
18
2.5. Weak Pareto Optimality
19
2.6. Value Function
21
2.7. Efficiency
23
2.8. From One Solution to Another
25
2.8.1. Trade-Offs
 26
2.8.2. Marginal Rate of Substitution
 27
2.9. Proper Pareto Optimality
29
2.10. Pareto Optimality Tests with Existence Results
 33
3. THEORETICAL BACKGROUND 37
3.1. Differentiable Optimality Conditions
 37
3.1.1. First-Order Conditions
37
3.1.2. Second-Order Conditions
42
3.1.3. Conditions for Proper Pareto Optimality
 43
3.2. Nondifferentiable Optimality Conditions
45
3.2.1. First-Order Conditions
47
3.2.2. Second-Order Conditions
 52
3.3. More Optimality Conditions
 54
3.4. Sensitivity Analysis and Duality
 56
Part II METHODS
1. INTRODUCTION 61
2. NO-PREFERENCE METHODS 67
2.1. Method of the Global Criterion
 67
2.1.1. Different Metrics
67
2.1.2. Theoretical Results
69
2.1.3. Concluding Remarks
 71
2.2. Multiobjective Proximal Bundle Method
 71
2.2.1. Introduction
71
2.2.2. MPB Algorithm
73
2.2.3. Theoretical Results
75
2.2.4. Concluding Remarks
75
3. A POSTERIORI METHODS 77
3.1. Weighting Method
78
3.1.1. Theoretical Results
78
3.1.2. Applications and Extensions
82
3.1.3. Weighting Method as an A Priori Method
 83
3.1.4. Concluding Remarks
84
3.2. e-Constraint Method
 85
3.2.1. Theoretical Results on Weak and Pareto Optimality
85
3.2.2. Connections with the Weighting Method
 88
3.2.3. Theoretical Results on Proper Pareto Optimality
89
3.2.4. Connections with Trade-Off Rates
 92
3.2.5. Applications and Extensions
94
3.2.6. Concluding Remarks
95
3.3. Hybrid Method
 96
3.4. Method of Weighted Metrics
 97
3.4.1. Introduction
 97
3.4.2. Theoretical Results
 98
3.4.3. Comments
99
3.4.4. Connections with Trade-Off Rates
100
3.4.5. Variants of the Weighted Tchebycheff Problem
 100
3.4.6. Connections with Global Trade-Offs
103
3.4.7. Applications and Extensions
106
3.4.8. Concluding Remarks
106
3.5. Achievement Scalarizing Function Approach
107
3.5.1. Introduction
107
3.5.2. Theoretical Results
108
3.5.3. Comments
110
3.5.4. Concluding Remarks
112
3.6. Other A Posteriori Methods
112
4. A PRIORI METHODS 115
4.1. Value Function Method
115
4.1.1. Introduction
115
4.1.2. Comments
116
4.1.3. Concluding Remarks
117
4.2. Lexicographic Ordering
118
4.2.1. Introduction
118
4.2.2. Comments
120
4.2.3. Concluding Remarks
120
4.3. Goal Programming
121
4.3.1. Introduction
121
4.3.2. Different Approaches
122
4.3.3. Comments
126
4.3.4. Applications and Extensions
127
4.3.5. Concluding Remarks
129
5. INTERACTIVE METHODS131
5.1. Interactive Surrogate Worth Trade-Off Method
 136
5.1.1. Introduction
136
5.1.2. ISWT Algorithm
137
5.1.3. Comments
140
5.1.4. Concluding Remarks
 141
5.2. Geoffrion-Dyer-Feinberg Method
141
5.2.1. Introduction
141
5.2.2. GDF Algorithm
143
5.2.3. Comments
146
5.2.4. Applications and Extensions
146
5.2.5. Concluding Remarks
 148
5.3. Sequential Proxy Optimization Technique
149
5.3.1. Introduction
149
5.3.2. SPOT Algorithm
151
5.3.3. Comments
152
5.3.4. Applications and Extensions
153
5.3.5. Concluding Remarks
153
5.4. Tchebycheff Method
154
5.4.1. Introduction
154
5.4.2. Tchebycheff Algorithm
158
5.4.3. Comments
159
5.4.4. Concluding Remarks
160
5.5. Step Method
161
5.5.1. Introduction
161
5.5.2. STEM Algorithm
162
5.5.3. Comments
163
5.5.4. Concluding Remarks
164
5.6. Reference Point Method
 164
5.6.1. Introduction
 165
5.6.2. Reference Point Algorithm
 165
5.6.3. Comments
 167
5.6.4. Implementation
 167
5.6.5. Applications and Extensions
169
5.6.6. Concluding Remarks
 170
5.7. GUESS Method
 170
5.7.1. Introduction
171
5.7.2. GUESS Algorithm
 173
5.7.3. Comments
173
5.7.4. Concluding Remarks
174
5.8. Satisficing Trade-Off Method
174
5.8.1. Introduction
174
5.8.2. STOM Algorithm
176
5.8.3. Comments
 177
5.8.4. Implementation
 178
5.8.5. Applications and Extensions
178
5.8.6. Concluding Remarks
179
5.9. Light Beam Search
179
5.9.1. Introduction
 180
5.9.2. Light Beam Algorithm
 182
5.9.3. Comments
183
5.9.4. Concluding Remarks
 184
5.10. Reference Direction Approach
184
5.10.1. Introduction
184
5.10.2. Reference Direction Approach Algorithm
185
5.10.3. Comments
187
5.10.4. Concluding Remarks
189
5.11. Reference Direction Method
 190
5.11.1. Introduction
190
5.11.2. RD Algorithm
 192
5.11.3. Comments
 193
5.11.4. Concluding Remarks
193
5.12. NIMBUS Method
195
5.12.1. Introduction
195
5.12.2. Vector Subproblem
 197
5.12.3. Scalar Subproblem
 198
5.12.4. NIMBUS Algorithm
198
5.12.5. Optimality Results
201
5.12.6. Comparison of the Two Versions
203
5.12.7. Comments
205
5.12.8. Implementations
205
5.12.9. Applications
206
5.12.10. Concluding Remarks
207
5.13. Other Interactive Methods
208
5.13.1. Methods Based on Goal Programming
208
5.13.2. Methods Based on Weighted Metrics
209
5.13.3. Methods Based on Reference Points
210
5.13.4. Methods Based on Miscellaneous Ideas
211
Part III RELATED ISSUES
1. COMPARING METHODS217
1.1. Comparative Table of Interactive Methods Presented
218
1.2. Comparisons Available in the Literature
 219
1.2.1. Introduction
220
1.2.2. Noninteractive Tests
221
1.2.3. Interactive Tests with Human Decision Makers
 221
1.2.4. Interactive Tests with Value Function
s225
1.2.5. Comparisons Based on Intuition
226
1.3. Selecting a Method
227
1.3.1. General Guidelines
227
1.3.2. Method Selection Tools
228
1.3.3. Decision Tree
229
2. SOFTWARE233
2.1. Introduction
233
2.2. Review
235
3. GRAPHICAL ILLUSTRATION239
3.1. Introduction
239
3.2. Illustrating the Pareto Optimal Set
240
3.3. Illustrating a Set of Alternatives
240
3.3.1. Value Path
240
3.3.2. Bar Chart
242
3.3.3. Star Coordinate System
243
3.3.4. Spider-Web Chart
243
3.3.5. Petal Diagram
244
3.3.6. Scatterplot Matrix
 245
3.3.7. Other Illustrative Means
246
3.3.8. General Remarks
247
4. FUTURE DIRECTIONS 251
5. EPILOGUE 255
REFERENCES 257
INDEX 293

Publication Information

Kaisa Miettinen: Nonlinear Multiobjective Optimization,
Kluwer Academic Publishers, 1999

Kluwer's International Series in Operations Research & Management Science,
Volume 12
ISBN 0-7923-8278-1

How to Order the Book

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Kaisa Miettinen, miettine@mit.jyu.fi