The tracial moment problem and trace-optimization of polynomials
type: Item
instance of: scholarly article
title: The tracial moment problem and trace-optimization of polynomials
zbMATH Open document ID: 1274.90256
author: Sabine Burgdorf
author: Kristijan Cafuta
author: Igor Klep
author: Janez Povh
published in: Mathematical Programming. Series A. Series B
publication date: 2013-03-18T00:00:00Z
review text: In this paper the authors consider the trace optimization of polynomials in non-commuting (nc) variables. To obtain a lower bound on the trace of such polynomials a relaxation is formulated in terms of semi-definite programming (SDP), which is based on sums of hermitian squares and commutators. It is shown that the lower bound is attained if the dual solution of the SDP satisfies a certain condition called flatness. Furthermore, the authors provide a procedure to determine the global minimizer of the trace. The paper is very well written and completed by numerous helpful examples. After the introduction and basic notations (nc polynomials, sum of hermitian squares), which are given in Section 1, the cyclic equivalence relation for nc polynomials, which considers two polynomials to be equal if their difference is a commutator, is introduced in Section 2. Based on this relation the convex cone \(\theta\) of nc polynomials that are cyclic equivalent to a sum of hermitian squares is stated. Using the Gram-Matrix method it is shown that each nc polynomial \(f\) is within the cone \(\theta\) if and only if it is cyclic equivalent to a product \(W^\ast G W\), where \(G\) is is a positive semi-definite matrix, \(W\) is a vector of all possible nc polynomials with degree less than half of the degree of \(f\) and \(W^\ast\) the corresponding vector of reversed nc polynomials. In Section 3, the SDP relaxation of the trace optimization problem is presented. It is shown that the relaxation yields a lower bound of the original problem and satisfies strong duality. Furthermore, the truncated tracial moment problem is given. It generalizes the tracial moment problem of Bayer and Teichmann and is used to formulated the flatness condition that guarantees exactness for the solution of the relaxation and the original problem. The flatness condition for the solution of the dual SDP and the corresponding truncated tracial sequence \(y\) of degree \(2k\) requires that the Hankel matrix \(…
reviewed by: Torsten Bosse
Mathematics Subject Classification ID: 90C22
Mathematics Subject Classification ID: 13J30
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 08B20
zbMATH DE Number: 6145980
zbMATH Keywords: Sum of squares
zbMATH Keywords: Non-commutative polynomial
zbMATH Keywords: Semi-definite programming
zbMATH Keywords: Tracial moment problem
zbMATH Keywords: Flat extension
zbMATH Keywords: Free positivity
zbMATH Keywords: Real algebraic geometry
describes a project that uses: Benchmarks for Optimization Software
describes a project that uses: SDPA
describes a project that uses: SparsePOP
describes a project that uses: GloptiPoly
describes a project that uses: NCSOStools
describes a project that uses: YALMIP
describes a project that uses: csminwel
describes a project that uses: Sostools
describes a project that uses: SeDuMi
describes a project that uses: SDPT3
describes a project that uses: NCAlgebra
MaRDI profile type: Publication
full work available at URL: s10107-011-0505-8
OpenAlex ID: W2064609151
cites work: Q5508534
cites work: The truncated tracial moment problem
cites work: Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics
cites work: The proof of Tchakaloff’s Theorem
cites work: Lectures on modern convex optimization. Analysis, algorithms, and engineering applications
cites work: Solution of the truncated complex moment problem for flat data
cites work: A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming
cites work: NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials
cites work: Q4698624
cites work: Classification of injective factors. Cases \(\mathrm{II}_1\), \(\mathrm{II}_\infty\), \(\mathrm{III}_\lambda\), \(\lambda\neq 1\)
cites work: Q3601985
cites work: Efficient decomposition of separable algebras.
cites work: ``Positive'' noncommutative polynomials are sums of squares.
cites work: Detecting Global Optimality and Extracting Solutions in GloptiPoly
cites work: GloptiPoly 3: moments, optimization and semidefinite programming
cites work: Semidefinite programming and sums of Hermitian squares of noncommutative polynomials
cites work: Connes' embedding conjecture and sums of Hermitian squares
cites work: Sums of Hermitian squares and the BMV conjecture
cites work: Q3999650
cites work: Global optimization with polynomials and the problem of moments
cites work: Q3601990
cites work: Factorization of operator-valued polynomials in several non-commuting variables
cites work: An independent benchmarking of SDP and SOCP solvers
cites work: A numerical algorithm for block-diagonal decomposition of matrix \(*\)-algebras with application to semidefinite programming
cites work: A numerical algorithm for block-diagonal decomposition of matrix \(*\)-algebras with general irreducible components
cites work: Regularization Methods for Semidefinite Programming
cites work: Q4324980
cites work: Semidefinite programming relaxations for semialgebraic problems
cites work: Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables
cites work: SOSTOOLS and Its Control Applications
cites work: Q4428719
cites work: An exact duality theory for semidefinite programming and its complexity implications
cites work: Solving the truncated moment problem solves the full moment problem
cites work: Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones
cites work: SDPT3 — A Matlab software package for semidefinite programming, Version 1.3
cites work: Handbook of semidefinite programming. Theory, algorithms, and applications
cites work: Implementation and evaluation of SDPA 6.0 (Semidefinite Programming Algorithm 6.0)
DOI: 10.1007/S10107-011-0505-8
description: scientific article; zbMATH DE number 6145980