Robust optimization is still a relatively new approach to optimization problems affected by uncertainty, but it has already proved so useful in real applications that it is difficult to tackle such problems today without considering this powerful methodology. Written by the principal developers of robust optimization, and describing the main achievements of a decade of research, this is the first book to provide a comprehensive and up-to-date account of the subject. Robust optimization is designed to meet some major challenges associated with uncertainty-affected optimization problems: to operate under lack of full information on the nature of uncertainty; to model the problem in a form that can be solved efficiently; and to provide guarantees about the performance of the solution. The book starts with a relatively simple treatment of uncertain linear programming, proceeding with a deep analysis of the interconnections between the construction of appropriate uncertainty sets and the classical chance constraints (probabilistic) approach. It then develops the robust optimization theory for uncertain conic quadratic and semidefinite optimization problems and dynamic (multistage) problems. The theory is supported by numerous examples and computational illustrations. An essential book for anyone working on optimization and decision making under uncertainty, Robust Optimization also makes an ideal graduate textbook on the subject.

Preface ix

PART I. ROBUST LINEAR OPTIMIZATION 1


Chapter 1. Uncertain Linear Optimization Problems and their Robust Counterparts 3
1.1 Data Uncertainty in Linear Optimization 3
1.2 Uncertain Linear Problems and their Robust Counterparts 7
1.3 Tractability of Robust Counterparts 16
1.4 Non-Affne Perturbations 23
1.5 Exercises 25
1.6 Notes and Remarks 25


Chapter 2. Robust Counterpart Approximations of Scalar Chance Constraints 27
2.1 How to Specify an Uncertainty Set 27
2.2 Chance Constraints and their Safe Tractable Approximations 28
2.3 Safe Tractable Approximations of Scalar Chance Constraints: Basic Examples 31
2.4 Extensions 44
2.5 Exercises 60
2.6 Notes and Remarks 64


Chapter 3. Globalized Robust Counterparts of Uncertain LO Problems 67
3.1 Globalized Robust Counterpart | Motivation and Definition 67
3.2 Computational Tractability of GRC 69
3.3 Example: Synthesis of Antenna Arrays 70
3.4 Exercises 79
3.5 Notes and Remarks 79


Chapter 4. More on Safe Tractable Approximations of Scalar Chance Constraints 81
4.1 Robust Counterpart Representation of a Safe Convex Approximation to a Scalar Chance Constraint 81
4.2 Bernstein Approximation of a Chance Constraint 83
4.3 From Bernstein Approximation to Conditional Value at Risk and Back 90
4.4 Majorization 105
4.5 Beyond the Case of Independent Linear Perturbations 109
4.6 Exercises 136
4.7 Notes and Remarks 145


PART II. ROBUST CONIC OPTIMIZATION 147


Chapter 5. Uncertain Conic Optimization: The Concepts 149
5.1 Uncertain Conic Optimization: Preliminaries 149
5.2 Robust Counterpart of Uncertain Conic Problem: Tractability 151
5.3 Safe Tractable Approximations of RCs of Uncertain Conic Inequalities 153
5.4 Exercises 156
5.5 Notes and Remarks 157


Chapter 6. Uncertain Conic Quadratic Problems with Tractable RCs 159
6.1 A Generic Solvable Case: Scenario Uncertainty 159
6.2 Solvable Case I: Simple Interval Uncertainty 160
6.3 Solvable Case II: Unstructured Norm-Bounded Uncertainty 161
6.4 Solvable Case III: Convex Quadratic Inequality with Un-structured Norm-Bounded Uncertainty 165
6.5 Solvable Case IV: CQI with Simple Ellipsoidal Uncertainty 167
6.6 Illustration: Robust Linear Estimation 173
6.7 Exercises 178
6.8 Notes and Remarks 178


Chapter 7. Approximating RCs of Uncertain Conic Quadratic Problems 179
7.1 Structured Norm-Bounded Uncertainty 179
7.2 The Case of -Ellipsoidal Uncertainty 195
7.3 Exercises 201
7.4 Notes and Remarks 201


Chapter 8. Uncertain Semidefinite Problems with Tractable RCs 203
8.1 Uncertain Semidefinite Problems 203
8.2 Tractability of RCs of Uncertain Semidefinite Problems 204
8.3 Exercises 222
8.4 Notes and Remarks 222


Chapter 9. Approximating RCs of Uncertain Semide¯nite
Problems 225
9.1 Tight Tractable Approximations of RCs of Uncertain SDPs
with Structured Norm-Bounded Uncertainty 225
9.2 Exercises 232
9.3 Notes and Remarks 234


Chapter 10. Approximating Chance Constrained CQIs and LMIs 235
10.1 Chance Constrained LMIs 235
10.2 The Approximation Scheme 240
10.3 Gaussian Majorization 252
10.4 Chance Constrained LMIs: Special Cases 255
10.5 Notes and Remarks 276


Chapter 11. Globalized Robust Counterparts of Uncertain Conic Problems 279
11.1 Globalized Robust Counterparts of Uncertain Conic Problems: De¯nition 279
11.2 Safe Tractable Approximations of GRCs 281
11.3 GRC of Uncertain Constraint: Decomposition 282
11.4 Tractability of GRCs 284
11.5 Illustration: Robust Analysis of Nonexpansive Dynamical Systems 292


Chapter 12. Robust Classification and Estimation 301
12.1 Robust Support Vector Machines 301
12.2 Robust Classification and Regression 309
12.3 Affine Uncertainty Models 325
12.4 Random Affine Uncertainty Models 331
12.5 Exercises 336
12.6 Notes and remarks 337


PART III. ROBUST MULTI-STAGE OPTIMIZATION 339


Chapter 13. Robust Markov Decision Processes 341
13.1 Markov Decision Processes 341
13.2 The Robust MDP Problems 345
13.3 The Robust Bellman Recursion on Finite Horizon 347
13.4 Notes and Remarks 352


Chapter 14. Robust Adjustable Multistage Optimization 355
14.1 Adjustable Robust Optimization: Motivation 355
14.2 Adjustable Robust Counterpart 357
14.3 Affinely Adjustable Robust Counterparts 368
14.4 Adjustable Robust Optimization and Synthesis of Linear Controllers 392
14.5 Exercises 408
14.6 Notes and Remarks 411


PART IV. SELECTED APPLICATIONS 415


Chapter 15. Selected Applications 417
15.1 Robust Linear Regression and Manufacturing of TV Tubes 417
15.2 Inventory Management with Flexible Commitment Contracts 421
15.3 Controlling a Multi-Echelon Multi-Period Supply Chain 432


Appendix A. Notation and Prerequisites 447
A.1 Notation 447
A.2 Conic Programming 448
A.3 Efficient Solvability of Convex Programming 460
Appendix B. Some Auxiliary Proofs 469
B.1 Proofs for Chapter 4 469
B.2 S-Lemma 481
B.3 Approximate S-Lemma 483
B.4 Matrix Cube Theorem 489
B.5 Proofs for Chapter 10 506
Appendix C. Solutions to Selected Exercises 511
C.1 Chapter 1 511
C.2 Chapter 2 511
C.3 Chapter 3 513
C.4 Chapter 4 513
C.5 Chapter 5 516
C.6 Chapter 6 519
C.7 Chapter 7 520
C.8 Chapter 8 521
C.9 Chapter 9 523
C.10 Chapter 12 525
C.11 Chapter 14 527


Bibliography 531
Index 539