The eld of arti cial intelligence (AI) and the law is on the cusp of a revolution that began with text analytic programs like IBM’s Watson and Debater and the open-source informa- tion management architectures on which they are based. Today, new legal applications are beginning to appear, and this book – designed to explain computational processes to non-programmers – describes how they will change the practice of law, speci cally by connecting computational models of legal reasoning directly with legal text, generat- ing arguments for and against particular outcomes, predicting outcomes, and explaining these predictions with reasons that legal professionals will be able to evaluate for them- selves. These legal apps will support conceptual legal information retrieval and enable cognitive computing, enabling a collaboration between humans and computers in which each performs the kinds of intelligent activities that they can do best. Anyone interested in how AI is changing the practice of law should read this illuminating work. Dr. Kevin D. Ashley is a Professor of Law and Intelligent Systems at the University of Pittsburgh, Senior Scientist, Learning Research and Development Center, and Adjunct Professor of Computer Science. He received a B.A. from Princeton University, a JD from Harvard Law School, and Ph.D. in computer science from the University of Mas- sachusetts. A visiting scientist at the IBM Thomas J. Watson Research Center, NSF Presidential Young Investigator, and Fellow of the American Association for Arti cial Intelligence, he is co-Editor-in-Chief of Arti cial Intelligence and Law and teaches in the University of Bologna Erasmus Mundus doctoral program in Law, Science, and Technology.

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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