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Table of 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
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