The Resource Optimization of polynomials in non-commuting variables, Sabine Burgdorf, Igor Klep, Janez Povh, (electronic book)
Optimization of polynomials in non-commuting variables, Sabine Burgdorf, Igor Klep, Janez Povh, (electronic book)
Resource Information
The item Optimization of polynomials in non-commuting 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 non-commuting 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 non-commuting variables. Researchers in non-commutative 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 non-commuting 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 Non-commuting Variables; 1.3 Sums of Hermitian Squares and Gram Matrices; 1.4 Quadratic Modules and Semialgebraic Sets; 1.5 Gelfand-Naimark-Segal'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 Trace-Positivity1.11 Wedderburn Theorem; 1.12 Curto-Fialkow'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 Non-commuting 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 Non-commutative 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 Non-commuting 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 non-commuting variables
- Title
- Optimization of polynomials in non-commuting 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 non-commuting variables. Researchers in non-commutative 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 non-commuting 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 non-commuting 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 Non-commuting Variables; 1.3 Sums of Hermitian Squares and Gram Matrices; 1.4 Quadratic Modules and Semialgebraic Sets; 1.5 Gelfand-Naimark-Segal'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 Trace-Positivity1.11 Wedderburn Theorem; 1.12 Curto-Fialkow'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 Non-commuting 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 Non-commutative 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 Non-commuting 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
- 10.1007/978-3-319-33338-0
- 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 non-commuting 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 Non-commuting Variables; 1.3 Sums of Hermitian Squares and Gram Matrices; 1.4 Quadratic Modules and Semialgebraic Sets; 1.5 Gelfand-Naimark-Segal'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 Trace-Positivity1.11 Wedderburn Theorem; 1.12 Curto-Fialkow'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 Non-commuting 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 Non-commutative 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 Non-commuting 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
- 10.1007/978-3-319-33338-0
- 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 fa-external-link-square fa-fw"></i> Data from <span resource="http://link.liverpool.ac.uk/portal/Optimization-of-polynomials-in-non-commuting/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/Optimization-of-polynomials-in-non-commuting/kQppyVyVoJ8/">Optimization of polynomials in non-commuting 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>
