The Resource Optimization of polynomials in noncommuting variables, Sabine Burgdorf, Igor Klep, Janez Povh, (electronic book)
Optimization of polynomials in noncommuting variables, Sabine Burgdorf, Igor Klep, Janez Povh, (electronic book)
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The item Optimization of polynomials in noncommuting variables, Sabine Burgdorf, Igor Klep, Janez Povh, (electronic book) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Liverpool.This item is available to borrow from 1 library branch.
Resource Information
The item Optimization of polynomials in noncommuting variables, Sabine Burgdorf, Igor Klep, Janez Povh, (electronic book) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Liverpool.
This item is available to borrow from 1 library branch.
 Summary
 This book presents recent results on positivity and optimization of polynomials in noncommuting variables. Researchers in noncommutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in noncommuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms
 Language
 eng
 Extent
 1 online resource (xv, 104 pages)
 Contents

 Introduction; Organization of the Book; References; Contents; List of Figures; List of Tables; 1 Selected Results from Algebra and Mathematical Optimization ; 1.1 Positive Semidefinite Matrices; 1.2 Words and Polynomials in Noncommuting Variables; 1.3 Sums of Hermitian Squares and Gram Matrices; 1.4 Quadratic Modules and Semialgebraic Sets; 1.5 GelfandNaimarkSegal's Construction; 1.6 Sums of Hermitian Squares and Positivity; 1.7 Vanishing Nc Polynomials; 1.8 Hankel Matrices and Flatness; 1.9 Commutators, Cyclic Equivalence, and Trace Zero Polynomials
 1.10 Cyclic Quadratic Modules and TracePositivity1.11 Wedderburn Theorem; 1.12 CurtoFialkow's Theorems; Implementation; 1.13 Semidefinite Programming; References; 2 Detecting Sums of Hermitian Squares; 2.1 Introduction; 2.2 The Gram Matrix Method; 2.3 Newton Chip Method; 2.4 Augmented Newton Chip Method; 2.5 Implementation; 2.5.1 On the Gram Matrix Method; 2.5.2 Software Package NCSOStools; References; 3 Cyclic Equivalence to Sums of Hermitian Squares; 3.1 Introduction; 3.2 The Cyclic Degree; 3.3 The Tracial Newton Polytope; 3.4 The Tracial Gram Matrix Method; 3.5 Implementation
 3.5.1 Detecting Members of Theta3.5.2 BMV Polynomials; References; 4 Eigenvalue Optimization of Polynomials in Noncommuting Variables; 4.1 Introduction; 4.2 Unconstrained Optimization; 4.2.1 Unconstrained Optimization as a Single SDP; 4.2.2 Extracting Optimizers for the Unconstrained Case; 4.3 Constrained Eigenvalue Optimization of Noncommutative Polynomials; 4.3.1 Approximation Hierarchy; 4.3.2 Extracting Optimizers; 4.4 Constrained Optimization over the Nc Ball and the Nc Polydisc; 4.4.1 Approximation Hierarchies Contain Only One Member; 4.4.2 Extracting Optimizers; 4.5 Implementation
 4.5.1 Application to Quantum MechanicsReferences; 5 Trace Optimization of Polynomials in Noncommuting Variables; 5.1 Introduction; 5.2 Unconstrained Trace Optimization; 5.3 Constrained Trace Optimization; 5.4 Flatness and Extracting Optimizers; 5.5 Implementation; References; List of Symbols; Index
 Isbn
 9783319333380
 Label
 Optimization of polynomials in noncommuting variables
 Title
 Optimization of polynomials in noncommuting variables
 Statement of responsibility
 Sabine Burgdorf, Igor Klep, Janez Povh
 Language
 eng
 Summary
 This book presents recent results on positivity and optimization of polynomials in noncommuting variables. Researchers in noncommutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in noncommuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms
 Cataloging source
 N$T
 http://library.link/vocab/creatorName
 Burgdorf, Sabine
 Dewey number
 519.6
 Illustrations
 illustrations
 Index
 index present
 LC call number
 QA402.5
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 http://library.link/vocab/relatedWorkOrContributorDate
 1973
 http://library.link/vocab/relatedWorkOrContributorName

 Klep, Igor
 Povh, Janez
 Series statement
 SpringerBriefs in mathematics,
 http://library.link/vocab/subjectName

 Mathematical optimization
 Polynomials
 Variables (Mathematics)
 Label
 Optimization of polynomials in noncommuting variables, Sabine Burgdorf, Igor Klep, Janez Povh, (electronic book)
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents

 Introduction; Organization of the Book; References; Contents; List of Figures; List of Tables; 1 Selected Results from Algebra and Mathematical Optimization ; 1.1 Positive Semidefinite Matrices; 1.2 Words and Polynomials in Noncommuting Variables; 1.3 Sums of Hermitian Squares and Gram Matrices; 1.4 Quadratic Modules and Semialgebraic Sets; 1.5 GelfandNaimarkSegal's Construction; 1.6 Sums of Hermitian Squares and Positivity; 1.7 Vanishing Nc Polynomials; 1.8 Hankel Matrices and Flatness; 1.9 Commutators, Cyclic Equivalence, and Trace Zero Polynomials
 1.10 Cyclic Quadratic Modules and TracePositivity1.11 Wedderburn Theorem; 1.12 CurtoFialkow's Theorems; Implementation; 1.13 Semidefinite Programming; References; 2 Detecting Sums of Hermitian Squares; 2.1 Introduction; 2.2 The Gram Matrix Method; 2.3 Newton Chip Method; 2.4 Augmented Newton Chip Method; 2.5 Implementation; 2.5.1 On the Gram Matrix Method; 2.5.2 Software Package NCSOStools; References; 3 Cyclic Equivalence to Sums of Hermitian Squares; 3.1 Introduction; 3.2 The Cyclic Degree; 3.3 The Tracial Newton Polytope; 3.4 The Tracial Gram Matrix Method; 3.5 Implementation
 3.5.1 Detecting Members of Theta3.5.2 BMV Polynomials; References; 4 Eigenvalue Optimization of Polynomials in Noncommuting Variables; 4.1 Introduction; 4.2 Unconstrained Optimization; 4.2.1 Unconstrained Optimization as a Single SDP; 4.2.2 Extracting Optimizers for the Unconstrained Case; 4.3 Constrained Eigenvalue Optimization of Noncommutative Polynomials; 4.3.1 Approximation Hierarchy; 4.3.2 Extracting Optimizers; 4.4 Constrained Optimization over the Nc Ball and the Nc Polydisc; 4.4.1 Approximation Hierarchies Contain Only One Member; 4.4.2 Extracting Optimizers; 4.5 Implementation
 4.5.1 Application to Quantum MechanicsReferences; 5 Trace Optimization of Polynomials in Noncommuting Variables; 5.1 Introduction; 5.2 Unconstrained Trace Optimization; 5.3 Constrained Trace Optimization; 5.4 Flatness and Extracting Optimizers; 5.5 Implementation; References; List of Symbols; Index
 Control code
 SPR951623796
 Dimensions
 unknown
 Extent
 1 online resource (xv, 104 pages)
 File format
 unknown
 Form of item
 online
 Isbn
 9783319333380
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other control number
 10.1007/9783319333380
 Other physical details
 color illustrations.
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 Label
 Optimization of polynomials in noncommuting variables, Sabine Burgdorf, Igor Klep, Janez Povh, (electronic book)
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents

 Introduction; Organization of the Book; References; Contents; List of Figures; List of Tables; 1 Selected Results from Algebra and Mathematical Optimization ; 1.1 Positive Semidefinite Matrices; 1.2 Words and Polynomials in Noncommuting Variables; 1.3 Sums of Hermitian Squares and Gram Matrices; 1.4 Quadratic Modules and Semialgebraic Sets; 1.5 GelfandNaimarkSegal's Construction; 1.6 Sums of Hermitian Squares and Positivity; 1.7 Vanishing Nc Polynomials; 1.8 Hankel Matrices and Flatness; 1.9 Commutators, Cyclic Equivalence, and Trace Zero Polynomials
 1.10 Cyclic Quadratic Modules and TracePositivity1.11 Wedderburn Theorem; 1.12 CurtoFialkow's Theorems; Implementation; 1.13 Semidefinite Programming; References; 2 Detecting Sums of Hermitian Squares; 2.1 Introduction; 2.2 The Gram Matrix Method; 2.3 Newton Chip Method; 2.4 Augmented Newton Chip Method; 2.5 Implementation; 2.5.1 On the Gram Matrix Method; 2.5.2 Software Package NCSOStools; References; 3 Cyclic Equivalence to Sums of Hermitian Squares; 3.1 Introduction; 3.2 The Cyclic Degree; 3.3 The Tracial Newton Polytope; 3.4 The Tracial Gram Matrix Method; 3.5 Implementation
 3.5.1 Detecting Members of Theta3.5.2 BMV Polynomials; References; 4 Eigenvalue Optimization of Polynomials in Noncommuting Variables; 4.1 Introduction; 4.2 Unconstrained Optimization; 4.2.1 Unconstrained Optimization as a Single SDP; 4.2.2 Extracting Optimizers for the Unconstrained Case; 4.3 Constrained Eigenvalue Optimization of Noncommutative Polynomials; 4.3.1 Approximation Hierarchy; 4.3.2 Extracting Optimizers; 4.4 Constrained Optimization over the Nc Ball and the Nc Polydisc; 4.4.1 Approximation Hierarchies Contain Only One Member; 4.4.2 Extracting Optimizers; 4.5 Implementation
 4.5.1 Application to Quantum MechanicsReferences; 5 Trace Optimization of Polynomials in Noncommuting Variables; 5.1 Introduction; 5.2 Unconstrained Trace Optimization; 5.3 Constrained Trace Optimization; 5.4 Flatness and Extracting Optimizers; 5.5 Implementation; References; List of Symbols; Index
 Control code
 SPR951623796
 Dimensions
 unknown
 Extent
 1 online resource (xv, 104 pages)
 File format
 unknown
 Form of item
 online
 Isbn
 9783319333380
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other control number
 10.1007/9783319333380
 Other physical details
 color illustrations.
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
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<div class="citation" vocab="http://schema.org/"><i class="fa faexternallinksquare fafw"></i> Data from <span resource="http://link.liverpool.ac.uk/portal/Optimizationofpolynomialsinnoncommuting/kQppyVyVoJ8/" typeof="Book http://bibfra.me/vocab/lite/Item"><span property="name http://bibfra.me/vocab/lite/label"><a href="http://link.liverpool.ac.uk/portal/Optimizationofpolynomialsinnoncommuting/kQppyVyVoJ8/">Optimization of polynomials in noncommuting variables, Sabine Burgdorf, Igor Klep, Janez Povh, (electronic book)</a></span>  <span property="potentialAction" typeOf="OrganizeAction"><span property="agent" typeof="LibrarySystem http://library.link/vocab/LibrarySystem" resource="http://link.liverpool.ac.uk/"><span property="name http://bibfra.me/vocab/lite/label"><a property="url" href="http://link.liverpool.ac.uk/">University of Liverpool</a></span></span></span></span></div>