  We consider optimization problems with polynomial inequality constraints in
non-commuting variables. These non-commuting variables are viewed as bounded
operators on a Hilbert space whose dimension is not fixed and the associated
polynomial inequalities as semidefinite positivity constraints. Such problems
arise naturally in quantum theory and quantum information science. To solve
them, we introduce a hierarchy of semidefinite programming relaxations which
generates a monotone sequence of lower bounds that converges to the optimal
solution. We also introduce a criterion to detect whether the global optimum is
reached at a given relaxation step and show how to extract a global optimizer
from the solution of the corresponding semidefinite programming problem.
