Optimization of Polynomials in Non-Commuting Variables
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Beschreibung
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.
Focuses on polynomial optimization problems in matrix unknowns Includes fundamental material from algebra, functional analysis and mathematical optimization Provides instructions on using NCSOStools open source package to obtain illustrated results Includes supplementary material: sn.pub/extras
Autor*in
Sabine Burgdorf
Themen in »Optimization of Polynomials in Non-Commuting Variables«
Newton chip method Newton cyclic chip method Sum of hermitian squares non-commutative algebraic geometry semidefinite programming polynomial data free analysis free real algebraic geometry quantum theory quantum information science mathematical optimization Unconstrained optimization quantum mechanics Extracting optimizers
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“The book covers the basics of NC polynomial optimization, building on elementary material from algebra and analysis as well as on more advanced concepts such as the Gelfand-Naimark-Segal construction. … This short yet very accessible book written by three leading experts in the field contains many examples, including explicit reproducible computations with their open source Mat lab toolbox NCSOS tools. It can be recommended to young researchers with an interest in polynomial optimization.” (Didier Henrion, Mathematical Reviews, May, 2017)
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