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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)

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
Creator
Contributor
Author
Subject
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
Member of
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)
Instantiates
Publication
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)
Publication
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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