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Sabine Burgdorf (sabine.burgdorfunikonstanz.de) Abstract: The main topic addressed in this paper is traceoptimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace f(A) can attain for a tuple of matrices A? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives effectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satisfies a certain condition called flatness, then the relaxation is exact. In this case it is shown how to extract global traceoptimizers with a procedure based on two ingredients. The first is the solution to the truncated tracial moment problem, and the other crucial component is the numerical implementation of the ArtinWedderburn theorem for matrix *algebras due to Murota, Kanno, Kojima, Kojima, and Maehara. Traceoptimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side  two topics with many applications, the most prominent being to linear systems engineering and quantum physics. The optimization problems discussed here facilitate new possibilities for applications, e.g. in operator algebras and statistical physics. Keywords: sum of squares, noncommutative polynomial, semidefinite programming, tracial moment problem, flat extension, free positivity, real algebraic geometry Category 1: Global Optimization Category 2: Linear, Cone and Semidefinite Programming (Semidefinite Programming ) Category 3: Optimization Software and Modeling Systems Citation: Mathematical Programming, 2013, vol. 137, iss. 12, pp. 557578 Download: [PDF] Entry Submitted: 04/19/2010 Modify/Update this entry  
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