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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Princeton, N.J
Princeton University Press
2009
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| Vydání: | 1 |
| Edice: | Princeton Series in Applied Mathematics |
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| ISBN: | 9780691143682, 0691143684, 9781400831050, 1400831059 |
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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 Optimizationalso 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. 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. 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 is difficult to tackle such problems today without considering this powerful methodology. The authors are the principal developers of 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 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. |
| Author | Nemirovski, Arkadi El Ghaoui, Laurent Ben-Tal, Aharon |
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| Keywords | Parametric family Identity matrix Inequality (mathematics) Stochastic Approximation algorithm Additive model Duality (optimization) Loss function Convex optimization Chaos theory Degeneracy (mathematics) Slack variable Best Accuracy and precision For All Practical Purposes 0O Strong duality With high probability Approximation Sensitivity analysis Decision problem Spherical model Almost surely Robust decision-making Decision rule Stochastic programming Robust control Curse of dimensionality Infimum and supremum Robust optimization Feasible region Big O notation Normal distribution Special case Stochastic optimization Mathematical optimization Time complexity Uncertainty Singular value Diagram (category theory) Maxima and minima Linear inequality Probability distribution Upper and lower bounds Spline (mathematics) Virtual displacement Markov chain Linear dynamical system Linear map Computational complexity theory Law of large numbers Linear matrix inequality Relative interior Theorem Dynamic programming Pairwise Coefficient Weak duality Probability worst and average case Linear programming Quantity Candidate solution Variable (mathematics) NP-hardness Simple set Central limit theorem Norm (mathematics) Optimal control Parameter Optimization problem |
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| Notes | Includes bibliography and index |
| OCLC | 439040007 |
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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 Big O notation Candidate solution Central limit theorem Chaos theory Coefficient Computational complexity theory Convex optimization Curse of dimensionality Decision problem Decision rule Degeneracy (mathematics) Diagram (category theory) Duality (optimization) Dynamic programming Feasible region For All Practical Purposes General Topics for Engineers Identity matrix Inequality (mathematics) Infimum and supremum Law of large numbers Linear dynamical system Linear inequality Linear map Linear matrix inequality Linear programming Loss function Markov chain Mathematical optimization MATHEMATICS MATHEMATICS / Applied MATHEMATICS / Linear & Nonlinear Programming Mathematische Optimierung Maxima and minima Norm (mathematics) Normal distribution NP-hardness Optimal control Optimization problem Pairwise Parameter Parametric family Probability Probability distribution Quantity Relative interior Robust control Robust decision-making Robust optimization Robustes Verfahren Sensitivity analysis Simple set Singular value Slack variable Special case Spherical model Spline (mathematics) Stochastic Stochastic optimization Stochastic programming Strong duality Theorem Theorie Time complexity Uncertainty Upper and lower bounds Variable (mathematics) Virtual displacement Weak duality With high probability worst and average case |
| 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 Chapter Fourteen. Robust Adjustable Multistage Optimization Appendix C: Solutions to Selected Exercises Chapter One. Uncertain Linear Optimization Problems and their Robust Counterparts Chapter Two. Robust Counterpart Approximations of Scalar Chance Constraints Chapter Eight. Uncertain Semidefinite Problems with Tractable RCs Part I. Robust Linear Optimization -- Part II. Robust Conic Optimization -- Index Chapter Twelve. Robust Classi¯cation and Estimation Part IV. Selected Applications -- Preface Chapter Ten. Approximating Chance Constrained CQIs and LMIs Appendix A: Notation and Prerequisites Chapter Five. Uncertain Conic Optimization: The Concepts Chapter Fifteen. Selected Applications Part III. Robust Multi-Stage Optimization -- - Chapter Six. Uncertain Conic Quadratic Problems with Tractable RCs / Chapter Three. Globalized Robust Counterparts of Uncertain LO Problems Appendix B: Some Auxiliary Proofs Contents Chapter Nine. Approximating RCs of Uncertain Semidefinite Problems Frontmatter -- Chapter Seven. Approximating RCs of Uncertain Conic Quadratic Problems Chapter Eleven. Globalized Robust Counterparts of Uncertain Conic Problems Chapter Thirteen. Robust Markov Decision Processes Bibliography |
| Title | Robust Optimization |
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| Volume | 28 |
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