Optimization of Polynomials in Non-Commuting Variables

1. Front Matter

   Pages i-xv

   PDF

2. Selected Results from Algebra and Mathematical Optimization

   • Sabine Burgdorf, Igor Klep, Janez Povh

   Pages 1-34

3. Detecting Sums of Hermitian Squares

   • Sabine Burgdorf, Igor Klep, Janez Povh

   Pages 35-44

4. Cyclic Equivalence to Sums of Hermitian Squares

   • Sabine Burgdorf, Igor Klep, Janez Povh

   Pages 45-62

5. Eigenvalue Optimization of Polynomials in Non-commuting Variables

   • Sabine Burgdorf, Igor Klep, Janez Povh

   Pages 63-85

6. Trace Optimization of Polynomials in Non-commuting Variables

   • Sabine Burgdorf, Igor Klep, Janez Povh

   Pages 87-99

7. Back Matter

   Pages 101-104

   PDF

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.

• 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

“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)

• Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

  Sabine Burgdorf

• Dept of Math, Bldg 303, Lvl 4, Rm 405, University of Auckland, Auckland, New Zealand

  Igor Klep

• in Novo Mesto, Faculty of Information Studies, Novo mesto, Slovenia

  Janez Povh

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