  This article focuses on optimization of polynomials in noncommuting
variables, while taking into account sparsity in the input data. A converging
hierarchy of semidefinite relaxations for eigenvalue and trace optimization is
provided. This hierarchy is a noncommutative analogue of results due to
Lasserre [SIAM J. Optim. 17(3) (2006), pp. 822--843] and Waki et al. [SIAM J.
Optim. 17(1) (2006), pp. 218--242]. The Gelfand-Naimark-Segal (GNS)
construction is applied to extract optimizers if flatness and irreducibility
conditions are satisfied. Among the main techniques used are amalgamation
results from operator algebra. The theoretical results are utilized to compute
lower bounds on minimal eigenvalue and trace of noncommutative polynomials from
the literature.
