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Robust Optimization

A. Ben-Tal, L. El Ghaoui, and A. Nemirovski
Princeton University Press
Publication Date: 
Number of Pages: 
Princeton Series in Applied Mathematics
[Reviewed by
Brian Borchers
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In introductory textbooks on optimization, it is typically assumed that all of the data in an optimization problem are known precisely. In actual practice, the data that enter into optimization problems are often uncertain. This uncertainty in problem coefficients can lead to situations in which a nominally optimal solution is practically infeasible or substantially suboptimal. Very early on, sensitivity analysis was developed as a way to determine how the optimal value of an optimization problem would change with respect to infinitesimal perturbations of the problem data. Later, stochastic programming approaches that assume probabilistic uncertainty in the problem data were developed. For example, in chance constrained optimization, the goal is to find a solution that is feasible and optimal with high probability, under assumptions about the probability distribution of the problem coefficients.

In robust optimization, we define a bounded “uncertainty set” of possible problem data, and then look for a solution that is guaranteed to be feasible under all possible variations of the problem data and that has the best objective value among these guaranteed feasible solutions. We could naively consider a version of the problem with an infinite number of constraints, one for each possible value of the problem data. The authors show how in many cases this robust counterpart to the original problem can be reformulated as a convex optimization problem with a finite set of constraints. The authors also discuss safe approximate reformulations of chance constrained stochastic programming problems as convex optimization problems. In various situations the robust counterpart to the uncertain problem might be formulated as a second order cone programming problem or a semidefinite programming problem. Recent advances in conic optimization have made the solution of these convex optimization problems a routine task.

In the first part of the book, the authors introduce the basic concepts of robust optimization and apply them to robust linear programming problems. In the second part of the book the approach is extended to robust versions of conic optimization problems including second order cone programs and semidefinite programming problems. In the third part of the book, the authors apply the robust optimization approach to multistage decision making problems.

The book contains a number of worked out examples that help to make this often theoretical material more concrete. The authors have also included numerous exercises throughout the book. Solutions to many of the exercises can be found in an appendix. A discussion of available software for solving robust optimization problems would have been extremely useful.

Robust optimization is an active area of research that is likely to find many practical applications in the future. This book is an authoritative reference that will be very useful to researchers working in this area. Furthermore, the book has been structured so that the first part could easily be used as the text for a graduate level course in robust optimization.

Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.

Preface ix


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


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


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


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