Optimization of Polynomials in Non-Commuting Variables

  • Sabine Burgdorf
  • Igor Klep
  • Janez Povh

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Sabine Burgdorf, Igor Klep, Janez Povh
    Pages 1-34
  3. Sabine Burgdorf, Igor Klep, Janez Povh
    Pages 35-44
  4. Sabine Burgdorf, Igor Klep, Janez Povh
    Pages 45-62
  5. Sabine Burgdorf, Igor Klep, Janez Povh
    Pages 63-85
  6. Sabine Burgdorf, Igor Klep, Janez Povh
    Pages 87-99
  7. Back Matter
    Pages 101-104

About this book

Introduction

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.

Keywords

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

Authors and affiliations

  • Sabine Burgdorf
    • 1
  • Igor Klep
    • 2
  • Janez Povh
    • 3
  1. 1.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands
  2. 2.Dept of Math, Bldg 303, Lvl 4, Rm 405University of AucklandAucklandNew Zealand
  3. 3.in Novo MestoFaculty of Information StudiesNovo mestoSlovenia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-33338-0
  • Copyright Information The Author(s) 2016
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-33336-6
  • Online ISBN 978-3-319-33338-0
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • About this book