Robust Optimization

Aharon Ben-Tal is professor of operations research at the Technion, Israel Institute for Technology. Laurent El Ghaoui is associate professor of electrical engineering and operations research at the University of California, Berkeley. Arkadi Nemirovski is professor of industrial and systems engineering at Georgia Institute of Technology.

"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, MAA Reviews "[T]his reference book gives an excellent and stimulating account of the classical and advanced results in the field, and should be consulted by all researchers and practitioners."--Joseph Frederic Bonnans, Zentralblatt MATH

• 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