Polynomial Optimization, Moments, and Applications / / Michal Kocvara, Bernard Mourrain, and Cordian Riener, editors

• Pubblicazione: Cham, Switzerland : , : Springer, , [2023]; ©2023
• Edizione: First edition.
• Descrizione fisica: 1 online resource (XIV, 266 p. 40 illus., 29 illus. in color.)
• Disciplina: 519.3
• Soggetto topico: Mathematical optimization; Polynomials
• Persona (resp. second.): KočvaraMichal; MourrainBernard; RienerCordian
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Acknowledgments -- Contents -- Contributors -- Polynomial Optimization, Certificates of Positivity, and Christoffel Function -- 1 Introduction -- 2 Notation, Definitions and Preliminary Results -- 3 The Moment-SOS Hierarchy in Polynomial Optimization -- 3.1 A Moment-SOS Hierarchy of Lower Bounds -- 3.2 A Moment-SOS Hierarchy of Upper Bounds -- 4 The Christoffel-Darboux Kernel and Christoffel Functions -- 4.1 Christoffel-Darboux Kernel -- 4.2 Christoffel Function -- 4.3 Some Distinguishing Properties of the CF -- 5 CF, Optimization, and SOS-Certificates of Positivity -- 5.1 The CF to Compare the Hierarchies of Upper and Lower Bounds -- 5.2 The CF and Positive Polynomials -- 5.3 A Disintegration of the CF -- 5.4 Positive Polynomials and Equilibrium Measure -- 6 Conclusion -- Appendix -- References -- Relative Entropy Methods in Constrained Polynomial and Signomial Optimization -- 1 Introduction -- 2 From Relative Entropy Programming to the SAGE Cone -- 2.1 Cones and Optimization -- 2.2 The Exponential Cone and the Relative Entropy Cone -- 2.3 The Basic AM/GM Idea -- 2.4 The SAGE Cone (Sums of Arithmetic-Geometric Exponentials) -- 3 Conditional Nonnegativity Over Convex Sets -- 4 The Circuit View for Unconstrained AM/GM Optimization -- 5 Sublinear Circuits -- 6 Irredundant Decompositions -- 7 Further Developments -- References -- Symmetries in Polynomial Optimization -- 1 Introduction -- 2 Preliminaries on the Moment-SOS Hierarchy in Polynomial Optimization and Semidefinite Programming -- 3 Using Representation Theory in SDPs for Sums-of-Squares -- 3.1 Basic Representation Theory -- 3.2 Representation Theory of Sn -- 3.3 Using Representation Theory to Simplify Semidefinite Formulations -- 4 Invariant Theory -- 4.1 Basics of Invariant Theory -- 4.2 Invariant Theory and Sums of Squares -- 4.3 Symmetric Sums of Squares. ; 5 Miscellaneous Approaches -- 5.1 Orbit Spaces and Polynomial Optimization -- 5.2 Reduction Via Orbit Decomposition -- 5.3 Symmetries of Optimizers -- References -- Copositive Matrices, Sums of Squares and the Stability Number of a Graph -- 1 Introduction -- 1.1 Organization of the Chapter -- 1.1.1 Notation -- 2 Preliminaries on Polynomial Optimization, Nonnegative Polynomials and Sums of Squares -- 2.1 Sum-of-Squares Certificates for Nonnegativity -- 2.2 Approximation Hierarchies for Polynomial Optimization -- 2.3 Optimality Conditions and Finite Convergence -- 3 Sum-of-Squares Approximations for COPn -- 3.1 Cones Based on Pólya's Nonnegativity Certificate -- 3.2 Lasserre-Type Approximation Cones -- 3.3 Links Between the Various Approximation Cones for COPn -- 4 Exactness of Sum-of-Squares Approximations for COPn -- 4.1 Exactness of the Conic Approximations Kn(r) -- 4.2 Exactness of the Conic Approximations LAS(r)n -- 4.3 The Cone of 55 Copositive Matrices -- 5 The Stability Number of a Graph α(G) -- 5.1 The Hierarchy ζ(r)(G) -- 5.2 The Hierarchy (r)(G) -- 5.2.1 Some Key Ingredients for the Proof for Theorem 22 -- 6 Concluding Remarks -- References -- Matrix Factorization Ranks Via Polynomial Optimization -- 1 Introduction and Motivation for Matrix Factorization Ranks -- 1.1 Applications of Nonnegative Factorization -- 1.2 Commonly Used Notation -- 1.3 On Computing the Nonnegative Rank -- 1.4 Other Factorization Ranks -- 2 Bounding Matrix Factorization Ranks -- 2.1 A Brief Introduction to Polynomial Optimization -- 2.2 Generalized Moment Problems -- 2.3 Constructing a Hierarchy of Lower Bounds for CP-Rank -- 2.4 A Note on Computing Hierarchies of SDPs -- 3 Exploiting Sparsity -- 3.1 An Abbreviated Introduction to Ideal Sparsity -- 3.2 Ideal Sparsity in Approximating CP-Rank -- 3.3 Advantages of the Sparse Hierarchy -- 4 Summary -- References. ; Polynomial Optimization in Geometric Modeling -- 1 Geometric Modeling and Polynomials -- 2 Polynomial Optimization Problems and Convex Relaxations -- 2.1 Sum of Squares Relaxations -- 2.2 Moment Relaxations -- 2.3 Computing the Minimizers -- 3 Minimal Enclosing Ellipsoids of Semi-algebraic Sets -- 4 Parameterized Surfaces -- 4.1 Closest Point and Surface-Surface Intersection -- 4.2 Bounding Box and Enclosing Ellipsoid of Parametric Surfaces -- 5 Robots and Mechanisms -- 5.1 Direct Kinematic Problem -- 5.2 Bounding Box of Robot Workspace -- 5.3 Enclosing Ellipsoid of Robot Workspace -- 5.4 Trajectories of a Parallel Robot -- 6 Conclusion -- References -- Assessing Safety for Control Systems Using Sum-of-Squares Programming -- 1 Introduction -- 1.1 Organization -- 1.2 Notation -- 2 Safety in Control Systems -- 2.1 Relationship Between Invariance and Safety -- 2.2 Control Invariance for Continuous-Time Systems -- 2.2.1 Control Invariance with Set Representation -- 2.2.2 Control Invariance with Function Representation -- 2.3 Control Invariance for Discrete-Time Systems -- 2.4 Summary -- 3 Sum-of-Squares Programming -- 3.1 Sum-of-Squares Decomposition -- 3.2 Convex Optimisation for Safety -- 3.3 Safety for Continuous-Time Systems -- 3.4 Safety for Discrete-Time Systems -- 3.5 Summary -- 4 Safety for Linear Systems with Constrained Inputs -- 4.1 Unit Peak Input -- 4.2 Summary -- 5 Applications -- 5.1 Nonlinear Control Affine System -- 5.2 Linear System -- 6 Conclusion -- Appendix -- References -- Polynomial Equations: Theory and Practice -- 1 Polynomial Equations in Optimization -- 2 Systems of Equations and Algebraic Varieties -- 3 Number of Solutions -- 3.1 Bézout's Theorem -- 3.2 Kushnirenko's Theorem -- 3.3 Bernstein's Theorem -- 4 Computational Methods -- 4.1 Normal Form Methods -- 4.2 Homotopy Continuation. ; 5 Case Study: 27 Lines on the Clebsch Surface -- References -- Index.
• Sommario/riassunto: Polynomial optimization is a fascinating field of study that has revolutionized the way we approach nonlinear problems described by polynomial constraints. The applications of this field range from production planning processes to transportation, energy consumption, and resource control. This introductory book explores the latest research developments in polynomial optimization, presenting the results of cutting-edge interdisciplinary work conducted by the European network POEMA. For the past four years, experts from various fields, including algebraists, geometers, computer scientists, and industrial actors, have collaborated in this network to create new methods that go beyond traditional paradigms of mathematical optimization. By exploiting new advances in algebra and convex geometry, these innovative approaches have resulted in significant scientific and technological advancements. This book aims to make these exciting developments accessible to a wider audience by gathering high-quality chapters on these hot topics. Aimed at both aspiring and established researchers, as well as industry professionals, this book will be an invaluable resource for anyone interested in polynomial optimization and its potential for real-world applications.
• Titolo autorizzato: Polynomial Optimization, Moments, and Applications
• ISBN: 3-031-38659-0
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910799488503321
• Lo trovi qui: Univ. Federico II
• Serie: Springer optimization and its applications ; ; Volume 206.