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We show that all the coefficients of the polynomial tr((A+ tB)) ∈ R[t] are nonnegative whenever m ≤ 13 is a nonnegative integer and A and B are positive semidefinite matrices of the same size. This has previously been known only for m ≤ 7. The validity of the statement for arbitrary m has recently been shown to be equivalent to the Bessis-Moussa-Villani… (More)

We show that Connes’ embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is nonnegative whenever self-adjoint contraction matrices of the same size are substituted for the variables. These algebraic… (More)

- Igor Klep, Markus Schweighofer
- Math. Oper. Res.
- 2013

Farkas’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the… (More)

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These… (More)

- J. William Helton, Igor Klep, Scott A. McCullough
- Math. Program.
- 2013

Given linear matrix inequalities (LMIs) L1 and L2 it is natural to ask: (Q1) when does one dominate the other, that is, does L1(X) 0 imply L2(X) 0? (Q2) when are they mutually dominant, that is, when do they have the same solution set? The matrix cube problem of Ben-Tal and Nemirovski [B-TN02] is an example of LMI domination. Hence such problems can be… (More)

- Kristijan Cafuta, Igor Klep, Janez Povh
- Optimization Methods and Software
- 2011

NCSOStools is a Matlab toolbox for • symbolic computation with polynomials in noncommuting variables; • constructing and solving sum of hermitian squares (with commutators) programs for polynomials in noncommuting variables. It can be used in combination with semidefinite programming software, such as SeDuMi, SDPA or SDPT3 to solve these constructed… (More)

- Igor Klep
- 2006

Let S ∪ {f} be a set of symmetric polynomials in noncommuting variables. If f satisfies a polynomial identity P i h ∗ i fhi = 1 + P i g ∗ i sigi for some si ∈ S ∪ {1}, then f is obviously nowhere negative semidefinite on the class of tuples of non-zero operators defined by the system of inequalities s ≥ 0 (s ∈ S). We prove the converse under the additional… (More)

- J. William Helton, Igor Klep, Scott A. McCullough, NICK SLINGLEND
- 2008

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These… (More)

- J. William Helton, Igor Klep, Raul Gomez
- SIAM J. Matrix Analysis Applications
- 2009

This paper treats two topics: matrices with sign patterns and Jacobians of certain mappings. The main topic is counting the number of plus and minus coefficients in the determinant expansion of sign patterns and of these Jacobians. The paper is motivated by an approach to chemical networks initiated by Craciun and Feinberg. We also give a graph-theoretic… (More)

- Igor Klep, Janez Povh
- 2009

An algorithm for finding sums of hermitian squares decompositions for polynomials in noncommuting variables is presented. The algorithm is based on the “Newton chip method”, a noncommutative analog of the classical Newton polytope method, and semidefinite programming.