Algorithmic aspects of sums of hermitian squares

Burgdorf, Sabine; Cafuta, Kristijan; Klep, Igor; Povh, Janez

Algorithmic aspects of sums of hermitian squares

Files

Burgdorf.pdf(437.55 KB)

Date

2012

Authors

Burgdorf, Sabine

Cafuta, Kristijan

Klep, Igor

Povh, Janez

Series

Konstanzer Schriften in Mathematik; 293

Publication type

Working Paper/Technical Report

Abstract

This paper presents an algorithm and its implementation in the software package NCSOStools for finding sums of hermitian squares and commutators decompositions for polynomials in noncommuting variables. The algorithm is based on noncommutative analogs of the classical Gram matrix method and the Newton polytope method, which allows us to use semidefinite programming. For rational polynomials numerical evidence can be tweaked to obtain an exact certificate using rational numbers. In the presence of Slater points, the Peyrl-Parrilo rounding and projecting method applies. On the other hand, in the absence of strict feasibility, a variant of the facial reduction is proposed to reduce the size of the semidefinite program and to enforce the existence of Slater points.

Subject (DDC)

510 Mathematics

Keywords

sum of squares,semidefinite programming,noncommutative polynomial,Matlab Toolbox,Newton polytope,free positivity

Cite This

ISO 690

BURGDORF, Sabine, Kristijan CAFUTA, Igor KLEP, Janez POVH, 2012. Algorithmic aspects of sums of hermitian squares

BibTex

@techreport{Burgdorf2012Algor-15338,
  year={2012},
  series={Konstanzer Schriften in Mathematik},
  title={Algorithmic aspects of sums of hermitian squares},
  number={293},
  author={Burgdorf, Sabine and Cafuta, Kristijan and Klep, Igor and Povh, Janez}
}

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