Optimization of polynomials in non-commuting variables. (2016)

• Record Type: Book
• Further Information: Note: Sabine Burgdorf, Igor Klep, Janez Povh.
• Authors: Burgdorf, Sabine; Klep, Igor; Povh, Janez, 1973-
• 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
• Publisher Details: Switzerland : Springer
• Publication Date: 2016
• Extent: 1 online resource (xv, 104 pages), color illustrations
• Subjects: 519.6; Mathematics; Mathematical optimization; Polynomials; Variables (Mathematics); MATHEMATICS / Applied; MATHEMATICS / Probability & Statistics / General; Mathematical optimization; Polynomials; Variables (Mathematics); Science -- Quantum Theory; Business & Economics -- Operations Research; Computers -- Mathematical & Statistical Software; Science -- System Theory; Mathematical theory of computation; Operational research; Mathematical & statistical software; Cybernetics & systems theory; Geometry, algebraic; Computer software; Systems theory; Mathematics -- Geometry -- Algebraic; Algebraic geometry; Electronic books; Electronic books
• Languages: English
• ISBNs: 9783319333380; 3319333380; 3319333364; 9783319333366
• Related ISBNs: 9783319333366
• Notes: Note: Includes bibliographical references and index.; Note: Online resource; title from PDF title page (SpringerLink, viewed June 20, 2016).
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.