Optimization of Polynomials in Non-Commuting Variables

Nonfiction, Science & Nature, Mathematics, Geometry, Science, Physics, Quantum Theory
Cover of the book Optimization of Polynomials in Non-Commuting Variables by Sabine Burgdorf, Igor Klep, Janez Povh, Springer International Publishing
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Author: Sabine Burgdorf, Igor Klep, Janez Povh ISBN: 9783319333380
Publisher: Springer International Publishing Publication: June 7, 2016
Imprint: Springer Language: English
Author: Sabine Burgdorf, Igor Klep, Janez Povh
ISBN: 9783319333380
Publisher: Springer International Publishing
Publication: June 7, 2016
Imprint: Springer
Language: English

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.

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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.

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