written by
Aharon Ben-Tal
Laurent El Ghaoui
Arkadi Nemirovski
Copyright © 2009 by Princeton University Press
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-Affine 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 Solv
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