Robust Optimization
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 r...
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| Main Authors | , , |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
Princeton, N.J
Princeton University Press
2009
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| Edition | 1 |
| Series | Princeton Series in Applied Mathematics |
| Subjects | |
| Online Access | Get full text |
| ISBN | 9780691143682 0691143684 9781400831050 1400831059 |
| DOI | 10.1515/9781400831050 |
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| Abstract | 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. |
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| AbstractList | 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. 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. Robust optimization is a fairly 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. The authors are the principal developers of robust optimization No detailed description available for "Robust Optimization". 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 Optimizationalso makes an ideal graduate textbook on the subject. |
| Author | Nemirovski, Arkadi El Ghaoui, Laurent Ben-Tal, Aharon |
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| Keywords | Parametric family Convex set Inequality (mathematics) Support vector machine Approximation algorithm Cumulative distribution function State variable Chaos theory Degeneracy (mathematics) Orientability Semi-infinite For All Practical Purposes 0O Approximation Decision problem Bifurcation theory Robust decision-making Stochastic programming Robust control Feasible region Robust optimization Normal distribution Stochastic optimization Best, worst and average case Mathematical optimization Time complexity Diagram (category theory) Linear inequality Free product Exponential function Spline (mathematics) Markov chain Linear map Law of large numbers Linear matrix inequality Theorem Dynamic programming P versus NP problem Weak duality Probability Linear programming Variable (mathematics) Simple set Central limit theorem Optimal control Skew-symmetric matrix Stochastic calculus Markov decision process Parameter Identity matrix Margin classifier Floor and ceiling functions Max-plus algebra Stochastic Additive model Duality (optimization) Loss function Convex optimization Slack variable Without loss of generality Accuracy and precision Random variable Strong duality With high probability Sensitivity analysis Ideal solution Spherical model Almost surely Decision rule Integer programming Convex hull Curse of dimensionality Infimum and supremum Proportionality (mathematics) Big O notation Stochastic control Special case Unimodality Uncertainty Singular value Maxima and minima Probability distribution Upper and lower bounds Virtual displacement Linear dynamical system Wiener filter Computational complexity theory Relative interior Likelihood-ratio test Pairwise Uniform distribution (discrete) Coefficient Multivariate normal distribution Linear regression Quantity Candidate solution Constrained optimization NP-hardness Norm (mathematics) Optimization problem |
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| Notes | Includes bibliography and index |
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| Snippet | Robust optimization is still a relatively new approach to optimization problems affected by uncertainty, but it has already proved so useful in real... No detailed description available for "Robust Optimization". Robust optimization is a fairly new approach to optimization problems affected by uncertainty, but it has already proved so useful in real applications that it... |
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| SubjectTerms | Accuracy and precision Additive model Almost surely Approximation Approximation algorithm Best, worst and average case Bifurcation theory Big O notation Candidate solution Central limit theorem Chaos theory Coefficient Computational complexity theory Constrained optimization Convex hull Convex optimization Convex set Cumulative distribution function Curse of dimensionality Decision problem Decision rule Degeneracy (mathematics) Diagram (category theory) Duality (optimization) Dynamic programming Exponential function Feasible region Floor and ceiling functions For All Practical Purposes Free product General Topics for Engineers Ideal solution Identity matrix Inequality (mathematics) Infimum and supremum Integer programming Law of large numbers Likelihood-ratio test Linear dynamical system Linear inequality Linear map Linear matrix inequality Linear programming Linear regression Loss function Margin classifier Markov chain Markov decision process Mathematical optimization MATHEMATICS MATHEMATICS / Linear & Nonlinear Programming Mathematische Optimierung Max-plus algebra Maxima and minima Multivariate normal distribution Norm (mathematics) Normal distribution NP-hardness Optimal control Optimization problem Orientability P versus NP problem Pairwise Parameter Parametric family Probability Probability distribution Proportionality (mathematics) Quantity Random variable Relative interior Robust control Robust decision-making Robust optimization Robustes Verfahren Semi-infinite Sensitivity analysis Simple set Singular value Skew-symmetric matrix Slack variable Special case Spherical model Spline (mathematics) State variable Stochastic Stochastic calculus Stochastic control Stochastic optimization Stochastic programming Strong duality Support vector machine Theorem Theorie Time complexity Uncertainty Uniform distribution (discrete) Unimodality Upper and lower bounds Variable (mathematics) Virtual displacement Weak duality Wiener filter With high probability Without loss of generality |
| SubjectTermsDisplay | Robust optimization |
| TableOfContents | Robust optimization -- Contents -- Preface -- Part I: Robust Linear Optimization -- Chapter One: Uncertain Linear Optimization Problems and their Robust Counterparts -- Chapter Two: Robust Counterpart Approximations of Scalar Chance Constraints -- Chapter Three: Globalized Robust Counterparts of Uncertain LO Problems -- Chapter Four: More on Safe Tractable Approximations of Scalar Chance Constraints -- Part II: Robust Conic Optimization -- Chapter Five: Uncertain Conic Optimization: The Concepts -- Chapter Six: Uncertain Conic Quadratic Problems with Tractable RCs -- Chapter Seven: Approximating RCs of Uncertain Conic Quadratic Problems -- Chapter Eight: Uncertain Semidefinite Problems with Tractable RCs -- Chapter Nine: Approximating RCs of Uncertain Semidefinite Problems -- Chapter Ten: Approximating Chance Constrained CQIs and LMIs -- Chapter Eleven: Globalized Robust Counterparts of Uncertain Conic Problems -- Chapter Twelve: Robust Classification and Estimation -- Part III: Robust Multi-Stage Optimization -- Chapter Thirteen: Robust Markov Decision Processes -- Chapter Fourteen: Robust Adjustable Multistage Optimization -- Part IV: Selected Applications -- Chapter Fifteen: Selected Applications -- Appendix A: Notation and Prerequisites -- Appendix B: Some Auxiliary Proofs -- Bibliography -- Index Front Matter Table of Contents Preface Chapter One: Uncertain Linear Optimization Problems and their Robust Counterparts Chapter Two: Robust Counterpart Approximations of Scalar Chance Constraints Chapter Three: Globalized Robust Counterparts of Uncertain LO Problems Chapter Four: More on Safe Tractable Approximations of Scalar Chance Constraints Chapter Five: Uncertain Conic Optimization: Chapter Six: Uncertain Conic Quadratic Problems with Tractable RCs Chapter Seven: Approximating RCs of Uncertain Conic Quadratic Problems Chapter Eight: Uncertain Semidefinite Problems with Tractable RCs Chapter Nine: Approximating RCs of Uncertain Semidefinite Problems Chapter Ten: Approximating Chance Constrained CQIs and LMIs Chapter Eleven: Globalized Robust Counterparts of Uncertain Conic Problems Chapter Twelve: Robust Classification and Estimation Chapter Thirteen: Robust Markov Decision Processes Chapter Fourteen: Robust Adjustable Multistage Optimization Chapter Fifteen: Selected Applications Appendix A. Appendix B. Appendix C. Bibliography Index Chapter 13. Robust Markov Decision Processes -- 13.1 Markov Decision Processes -- 13.2 The Robust MDP Problems -- 13.3 The Robust Bellman Recursion on Finite Horizon -- 13.4 Notes and Remarks -- Chapter 14. Robust Adjustable Multistage Optimization -- 14.1 Adjustable Robust Optimization: Motivation -- 14.2 Adjustable Robust Counterpart -- 14.3 Affinely Adjustable Robust Counterparts -- 14.4 Adjustable Robust Optimization and Synthesis of Linear Controllers -- 14.5 Exercises -- 14.6 Notes and Remarks -- PART IV. SELECTED APPLICATIONS -- Chapter 15. Selected Applications -- 15.1 Robust Linear Regression and Manufacturing of TV Tubes -- 15.2 Inventory Management with Flexible Commitment Contracts -- 15.3 Controlling a Multi-Echelon Multi-Period Supply Chain -- Appendix A. Notation and Prerequisites -- A.1 Notation -- A.2 Conic Programming -- A.3 Efficient Solvability of Convex Programming -- Appendix B. Some Auxiliary Proofs -- B.1 Proofs for Chapter 4 -- B.2 S-Lemma -- B.3 Approximate S-Lemma -- B.4 Matrix Cube Theorem -- B.5 Proofs for Chapter 1 -- Appendix C. Solutions to Selected Exercises -- C.1 Chapter 1 -- C.2 Chapter 2 -- C.3 Chapter 3 -- C.4 Chapter 4 -- C.5 Chapter 5 -- C.6 Chapter 6 -- C.7 Chapter 7 -- C.8 Chapter 8 -- C.9 Chapter 9 -- C.10 Chapter 12 -- C.11 Chapter 14 -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- I -- L -- M -- N -- P -- R -- S -- T -- U -- W Cover -- Title -- Copyright -- Contents -- Preface -- PART I. ROBUST LINEAR OPTIMIZATION -- Chapter 1. Uncertain Linear Optimization Problems and their Robust Counterparts -- 1.1 Data Uncertainty in Linear Optimization -- 1.2 Uncertain Linear Problems and their Robust Counterparts -- 1.3 Tractability of Robust Counterparts -- 1.4 Non-Affine Perturbations -- 1.5 Exercises -- 1.6 Notes and Remarks -- Chapter 2. Robust Counterpart Approximations of Scalar Chance Constraints -- 2.1 How to Specify an Uncertainty Set -- 2.2 Chance Constraints and their Safe Tractable Approximations -- 2.3 Safe Tractable Approximations of Scalar Chance Constraints: Basic Examples -- 2.4 Extensions -- 2.5 Exercises -- 2.6 Notes and Remarks -- Chapter 3. Globalized Robust Counterparts of Uncertain LO Problems -- 3.1 Globalized Robust Counterpart-Motivation and Definition -- 3.2 Computational Tractability of GRC -- 3.3 Example: Synthesis of Antenna Arrays -- 3.4 Exercises -- 3.5 Notes and Remarks -- Chapter 4. More on Safe Tractable Approximations of Scalar Chance Constraints -- 4.1 Robust Counterpart Representation of a Safe Convex Approximation to a Scalar Chance Constraint -- 4.2 Bernstein Approximation of a Chance Constraint -- 4.3 From Bernstein Approximation to Conditional Value at Risk and Back -- 4.4 Majorization -- 4.5 Beyond the Case of Independent Linear Perturbations -- 4.6 Exercises -- 4.7 Notes and Remarks -- PART II. ROBUST CONIC OPTIMIZATION -- Chapter 5. Uncertain Conic Optimization: The Concepts -- 5.1 Uncertain Conic Optimization: Preliminaries -- 5.2 Robust Counterpart of Uncertain Conic Problem: Tractability -- 5.3 Safe Tractable Approximations of RCs of Uncertain Conic Inequalities -- 5.4 Exercises -- 5.5 Notes and Remarks -- Chapter 6. Uncertain Conic Quadratic Problems with Tractable RCs -- 6.1 A Generic Solvable Case: Scenario Uncertainty 6.2 Solvable Case I: Simple Interval Uncertainty -- 6.3 Solvable Case II: Unstructured Norm-Bounded Uncertainty -- 6.4 Solvable Case III: Convex Quadratic Inequality with Unstructured Norm-Bounded Uncertainty -- 6.5 Solvable Case IV: CQI with Simple Ellipsoidal Uncertainty -- 6.6 Illustration: Robust Linear Estimation -- 6.7 Exercises -- 6.8 Notes and Remarks -- Chapter 7. Approximating RCs of Uncertain Conic Quadratic Problems -- 7.1 Structured Norm-Bounded Uncertainty -- 7.2 The Case of & -- #8745 -- -Ellipsoidal Uncertainty -- 7.3 Exercises -- 7.4 Notes and Remarks -- Chapter 8. Uncertain Semidefinite Problems with Tractable RCs -- 8.1 Uncertain Semidefinite Problems -- 8.2 Tractability of RCs of Uncertain Semidefinite Problems -- 8.3 Exercises -- 8.4 Notes and Remarks -- Chapter 9. Approximating RCs of Uncertain Semidefinite Problems -- 9.1 Tight Tractable Approximations of RCs of Uncertain SDPs with Structured Norm-Bounded Uncertainty -- 9.2 Exercises -- 9.3 Notes and Remarks -- Chapter 10. Approximating Chance Constrained CQIs and LMIs -- 10.1 Chance Constrained LMIs -- 10.2 The Approximation Scheme -- 10.3 Gaussian Majorization -- 10.4 Chance Constrained LMIs: Special Cases -- 10.5 Notes and Remarks -- Chapter 11. Globalized Robust Counterparts of Uncertain Conic Problems -- 11.1 Globalized Robust Counterparts of Uncertain Conic Problems: Definition -- 11.2 Safe Tractable Approximations of GRCs -- 11.3 GRC of Uncertain Constraint: Decomposition -- 11.4 Tractability of GRCs -- 11.5 Illustration: Robust Analysis of Nonexpansive Dynamical Systems -- Chapter 12. Robust Classification and Estimation -- 12.1 Robust Support Vector Machines -- 12.2 Robust Classification and Regression -- 12.3 Affine Uncertainty Models -- 12.4 Random Affine Uncertainty Models -- 12.5 Exercises -- 12.6 Notes and remarks -- PART III. ROBUST MULTI-STAGE OPTIMIZATION Chapter Four. More on Safe Tractable Approximations of Scalar Chance Constraints -- Contents -- Preface -- Appendix A: Notation and Prerequisites -- Appendix C: Solutions to Selected Exercises -- Part I. Robust Linear Optimization -- Part II. Robust Conic Optimization -- Index Chapter Ten. Approximating Chance Constrained CQIs and LMIs -- Chapter Two. Robust Counterpart Approximations of Scalar Chance Constraints -- Part IV. Selected Applications -- Chapter Seven. Approximating RCs of Uncertain Conic Quadratic Problems -- Chapter Three. Globalized Robust Counterparts of Uncertain LO Problems -- Chapter Nine. Approximating RCs of Uncertain Semidefinite Problems -- Bibliography -- Chapter Eleven. Globalized Robust Counterparts of Uncertain Conic Problems -- Chapter Fourteen. Robust Adjustable Multistage Optimization -- Chapter Eight. Uncertain Semidefinite Problems with Tractable RCs -- Chapter Five. Uncertain Conic Optimization: The Concepts -- Chapter Six. Uncertain Conic Quadratic Problems with Tractable RCs -- Part III. Robust Multi-Stage Optimization -- Appendix B: Some Auxiliary Proofs -- Chapter Thirteen. Robust Markov Decision Processes -- Frontmatter -- Chapter One. Uncertain Linear Optimization Problems and their Robust Counterparts -- Chapter Twelve. Robust Classi¯cation and Estimation -- Chapter Fifteen. Selected Applications -- |
| Title | Robust Optimization |
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