Self-Regularity A New Paradigm for Primal-Dual Interior-Point Algorithms

Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap b...

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Hlavní autori:	Peng, Jiming, Roos, Cornelis, Terlaky, Tamás
Médium:	E-kniha Kniha
Jazyk:	English
Vydavateľské údaje:	Princeton, N.J. ; Chichester Princeton University Press 2009
Vydanie:	1
Edícia:	Princeton series in applied mathematics
Predmet:	Algorithm Analysis of algorithms Analytic function Associative property Barrier function Block matrix Combinatorial optimization Combinatorics Complexity Conic optimization Continuous optimization Control theory Convex optimization Delft University of Technology Derivative Duality gap Eigenvalues and eigenvectors Embedding Equation Estimation Explanation Feasible region Filter design Function (mathematics) General Topics for Engineers Implementation Instance (computer science) Invertible matrix Iteration Jacobian matrix and determinant Jordan algebra Karush–Kuhn–Tucker conditions Linear complementarity problem Linear programming Lipschitz continuity Local convergence Loss function Mathematical optimization MATHEMATICS MATHEMATICS / Applied MATHEMATICS / Optimization Matrix function McMaster University Multiplication operator Newton's method Nonlinear programming Optimal control Optimization problem Parameter Pattern recognition Polynomial Programming (Mathematics) Quadratic function Requirement Result Scientific notation Self-concordant function Sensitivity analysis Simplex algorithm Simultaneous equations Singular value Smoothness Solution set Solver Special case Theorem Theory Time complexity Variable (mathematics) Variational inequality Variational principle Without loss of generality Yurii Nesterov Polynomial Technical report McMaster University Yurii Nesterov Continuous optimization Delft University of Technology Explanation Self-concordant function Conic optimization Result Complexity Loss function Convex optimization Variational inequality Eigenvalues and eigenvectors Invertible matrix Nonlinear programming Without loss of generality Combinatorics Duality gap Sensitivity analysis Estimation Pattern recognition Filter design Algorithm Matrix function Function (mathematics) Requirement Feasible region Karush–Kuhn–Tucker conditions Derivative Special case Analysis of algorithms Mathematical optimization Time complexity Simplex algorithm Quadratic function Solver Monograph Theory Barrier function Singular value Embedding Iteration Combinatorial optimization Implementation Lipschitz continuity Solution set Associative property Theorem Local convergence Block matrix Jacobian matrix and determinant Linear programming Equation Analytic function Linear complementarity problem Jordan algebra Smoothness Instance (computer science) Variable (mathematics) Simultaneous equations Optimal control Multiplication operator Variational principle Newton's method Parameter Scientific notation Control theory Optimization problem
ISBN:	9780691091921, 0691091927, 0691091935, 9780691091938, 9781400825134, 140082513X
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Abstract	Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap between the theory and practical performance of these two strategies. This book comes close to bridging that gap, presenting a new framework for the theory of primal-dual IPMs based on the notion of the self-regularity of a function. The authors deal with linear optimization, nonlinear complementarity problems, semidefinite optimization, and second-order conic optimization problems. The framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered can be interpreted as a path-following method or a potential reduction method. Starting from a primal-dual strictly feasible point, the algorithm chooses a search direction defined by some Newton-type system derived from the self-regular proximity. The iterate is then updated, with the iterates staying in a certain neighborhood of the central path until an approximate solution to the problem is found. By extensively exploring some intriguing properties of self-regular functions, the authors establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs. Researchers and postgraduate students in all areas of linear and nonlinear optimization will find this book an important and invaluable aid to their work.
AbstractList	Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap between the theory and practical performance of these two strategies. This book comes close to bridging that gap, presenting a new framework for the theory of primal-dual IPMs based on the notion of the self-regularity of a function. The authors deal with linear optimization, nonlinear complementarity problems, semidefinite optimization, and second-order conic optimization problems. The framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered can be interpreted as a path-following method or a potential reduction method. Starting from a primal-dual strictly feasible point, the algorithm chooses a search direction defined by some Newton-type system derived from the self-regular proximity. The iterate is then updated, with the iterates staying in a certain neighborhood of the central path until an approximate solution to the problem is found. By extensively exploring some intriguing properties of self-regular functions, the authors establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs. Researchers and postgraduate students in all areas of linear and nonlinear optimization will find this book an important and invaluable aid to their work. Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap between the theory and practical performance of these two strategies. This book comes close to bridging that gap, presenting a new framework for the theory of primal-dual IPMs based on the notion of the self-regularity of a function. The authors deal with linear optimization, nonlinear complementarity problems, semidefinite optimization, and second-order conic optimization problems. The framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered can be interpreted as a path-following method or a potential reduction method. Starting from a primal-dual strictly feasible point, the algorithm chooses a search direction defined by some Newton-type system derived from the self-regular proximity. The iterate is then updated, with the iterates staying in a certain neighborhood of the central path until an approximate solution to the problem is found. By extensively exploring some intriguing properties of self-regular functions, the authors establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs. Researchers and postgraduate students in all areas of linear and nonlinear optimization will find this book an important and invaluable aid to their work. Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap between the theory and practical performance of these two strategies. This book comes close to bridging that gap, presenting a new framework for the theory of primal-dual IPMs based on the notion of the self-regularity of a function. The authors deal with linear optimization, nonlinear complementarity problems, semidefinite optimization, and second-order conic optimization problems. The framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered can be interpreted as a path-following method or a potential reduction method. Starting from a primal-dual strictly feasible point, the algorithm chooses a search direction defined by some Newton-type system derived from the self-regular proximity. The iterate is then updated, with the iterates staying in a certain neighborhood of the central path until an approximate solution to the problem is found. By extensively exploring some intriguing properties of self-regular functions, the authors establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs. No detailed description available for "Self-Regularity".
Author	Terlaky, Tamás Roos, Cornelis Peng, Jiming
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Keywords	Polynomial Technical report McMaster University Yurii Nesterov Continuous optimization Delft University of Technology Explanation Self-concordant function Conic optimization Result Complexity Loss function Convex optimization Variational inequality Eigenvalues and eigenvectors Invertible matrix Nonlinear programming Without loss of generality Combinatorics Duality gap Sensitivity analysis Estimation Pattern recognition Filter design Algorithm Matrix function Function (mathematics) Requirement Feasible region Karush–Kuhn–Tucker conditions Derivative Special case Analysis of algorithms Mathematical optimization Time complexity Simplex algorithm Quadratic function Solver Monograph Theory Barrier function Singular value Embedding Iteration Combinatorial optimization Implementation Lipschitz continuity Solution set Associative property Theorem Local convergence Block matrix Jacobian matrix and determinant Linear programming Equation Analytic function Linear complementarity problem Jordan algebra Smoothness Instance (computer science) Variable (mathematics) Simultaneous equations Optimal control Multiplication operator Variational principle Newton's method Parameter Scientific notation Control theory Optimization problem
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SubjectTerms	Algorithm Analysis of algorithms Analytic function Associative property Barrier function Block matrix Combinatorial optimization Combinatorics Complexity Conic optimization Continuous optimization Control theory Convex optimization Delft University of Technology Derivative Duality gap Eigenvalues and eigenvectors Embedding Equation Estimation Explanation Feasible region Filter design Function (mathematics) General Topics for Engineers Implementation Instance (computer science) Invertible matrix Iteration Jacobian matrix and determinant Jordan algebra Karush–Kuhn–Tucker conditions Linear complementarity problem Linear programming Lipschitz continuity Local convergence Loss function Mathematical optimization MATHEMATICS MATHEMATICS / Applied MATHEMATICS / Optimization Matrix function McMaster University Multiplication operator Newton's method Nonlinear programming Optimal control Optimization problem Parameter Pattern recognition Polynomial Programming (Mathematics) Quadratic function Requirement Result Scientific notation Self-concordant function Sensitivity analysis Simplex algorithm Simultaneous equations Singular value Smoothness Solution set Solver Special case Theorem Theory Time complexity Variable (mathematics) Variational inequality Variational principle Without loss of generality Yurii Nesterov
SubjectTermsDisplay	Mathematical optimization
Subtitle	A New Paradigm for Primal-Dual Interior-Point Algorithms
TableOfContents	Self-regularity:A new paradigm for primal-dual interior-point algorithms -- Contents -- Preface -- Acknowledgements -- Notation -- List of Abbreviations -- Chapter 1: Introduction and Preliminaries -- Chapter 2: Self-Regular Functions and Their Properties -- Chapter 3: Primal-Dual Algorithms for Linear Optimization Based on Self-Regular Proximities -- Chapter 4: Interior-Point Methods for Complementarity Problems Based on Self-Regular Proximities -- Chapter 5: Primal-Dual Interior-Point Methods for Semidefinite Optimization Based on Self-Regular Proximities -- Chapter 6: Primal-Dual Interior-Point Methods for Second-Order Conic Optimization Based on Self-Regular Proximities -- Chapter 7: Initialization: Embedding Models for Linear Optimization, Complementarity Problems, Semidefinite Optimization and Second-Order Conic Optimization -- Chapter 8: Conclusions -- References -- Index Front Matter Table of Contents Preface Acknowledgments Notation List of Abbreviations Chapter 1: Introduction and Preliminaries Chapter 2: Self-Regular Functions and Their Properties Chapter 3: Primal-Dual Algorithms for Linear Optimization Based on Self-Regular Proximities Chapter 4: Interior-Point Methods for Complementarity Problems Based on Self-Regular Proximities Chapter 5: Primal-Dual Interior-Point Methods for Semidefinite Optimization Based on Self-Regular Proximities Chapter 6: Primal-Dual Interior-Point Methods for Second-Order Conic Optimization Based on Self-Regular Proximities Chapter 7: Initialization: Chapter 8: Conclusions References Index 5.3 New Search Directions for SDO -- 5.3.1 Scaling Schemes for SDO -- 5.3.2 Intermezzo: A Variational Principle for Scaling -- 5.3.3 New Proximities and Search Directions for SDO -- 5.4 New Polynomial Primal-Dual IPMs for SDO -- 5.4.1 The Algorithm -- 5.4.2 Complexity of the Algorithm -- Chapter 6. Primal-Dual Interior-Point Methods for Second-Order Conic Optimization Based on Self-Regular Proximities -- 6.1 Introduction to SOCO, Duality Theory and The Central Path -- 6.2 Preliminary Results on Functions Associated with Second-Order Cones -- 6.2.1 Jordan Algebra, Trace and Determinant -- 6.2.2 Functions and Derivatives Associated with Second-Order Cones -- 6.3 New Search Directions for SOCO -- 6.3.1 Preliminaries -- 6.3.2 Scaling Schemes for SOCO -- 6.3.3 Intermezzo: A Variational Principle for Scaling -- 6.3.4 New Proximities and Search Directions for SOCO -- 6.4 New IPMs for SOCO -- 6.4.1 The Algorithm -- 6.4.2 Complexity of the Algorithm -- Chapter 7. Initialization: Embedding Models for Linear Optimization, Complementarity Problems, Semidefinite Optimization and Second-Order Conic Optimization -- 7.1 The Self-Dual Embedding Model for LO -- 7.2 The Embedding Model for CP -- 7.3 Self-Dual Embedding Models for SDO and SOCO -- Chapter 8. Conclusions -- 8.1 A Survey of the Results and Future Research Topics -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- R -- S -- T -- V -- W -- Y -- Z Intro -- Contents -- Preface -- Acknowledgements -- Notation -- List of Abbreviations -- Chapter 1. Introduction and Preliminaries -- 1.1 Historical Background of Interior-Point Methods -- 1.1.1 Prelude -- 1.1.2 A Brief Review of Modern Interior-Point Methods -- 1.2 Primal-Dual Path-Following Algorithm for LO -- 1.2.1 Primal-Dual Model for LO, Duality Theory and the Central Path -- 1.2.2 Primal-Dual Newton Method for LO -- 1.2.3 Strategies in Path-following Algorithms and Motivation -- 1.3 Preliminaries and Scope of the Monograph -- 1.3.1 Preliminary Technical Results -- 1.3.2 Relation Between Proximities and Search Directions -- 1.3.3 Contents and Notational Abbreviations -- Chapter 2. Self-Regular Functions and Their Properties -- 2.1 An Introduction to Univariate Self-Regular Functions -- 2.2 Basic Properties of Univariate Self-Regular Functions -- 2.3 Relations Between S-R and S-C Functions -- Chapter 3. Primal-Dual Algorithms for Linear Optimization Based on Self-Regular Proximities -- 3.1 Self-Regular Functions in Rn++ and Self-Regular Proximities for LO -- 3.2 The Algorithm -- 3.3 Estimate of the Proximity After a Newton Step -- 3.4 Complexity of the Algorithm -- 3.5 Relaxing the Requirement on the Proximity Function -- Chapter 4. Interior-Point Methods for Complementarity Problems Based on Self-Regular Proximities -- 4.1 Introduction to CPs and the Central Path -- 4.2 Preliminary Results on P(k) Mappings -- 4.3 New Search Directions for P(k) CPs -- 4.4 Complexity of the Algorithm -- 4.4.1 Ingredients for Estimating the Proximity -- 4.4.2 Estimate of the Proximity After a Step -- 4.4.3 Complexity of the Algorithm for CPs -- Chapter 5. Primal-Dual Interior-Point Methods for Semidefinite Optimization Based on Self-Regular Proximities -- 5.1 Introduction to SDO, Duality Theory and Central Path -- 5.2 Preliminary Results on Matrix Functions Chapter 4. Interior-Point Methods for Complementarity Problems Based on Self- Regular Proximities Chapter 5. Primal-Dual Interior-Point Methods for Semidefinite Optimization Based on Self-Regular Proximities Chapter 1. Introduction and Preliminaries Index Chapter 2. Self-Regular Functions and Their Properties Chapter 6. Primal-Dual Interior-Point Methods for Second-Order Conic Optimization Based on Self-Regular Proximities Notation - / Contents Acknowledgments List of Abbreviations References Frontmatter -- Chapter 3. Primal-Dual Algorithms for Linear Optimization Based on Self-Regular Proximities Preface Chapter 7. Initialization: Embedding Models for Linear Optimization, Complementarity Problems, Semidefinite Optimization and Second-Order Conic Optimization Chapter 8. Conclusions
Title	Self-Regularity
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