We consider the problem of optimizing the state average of a polynomial of non-commuting variables, over all states and operators satisfying a number of polynomial constraints, and over all Hilbert spaces where such states and operators are defined. Such non-commutative polynomial optimization (NPO) problems are routinely solved through hierarchies of semidefinite programming (SDP) relaxations. By phrasing the general NPO problem in Lagrangian form, we heuristically derive, via small variations on the problem variables, state and operator optimality conditions, both of which can be enforced by adding new positive semidefinite constraints to the SDP hierarchies. State optimality conditions are satisfied by all Archimedean (that is, bounded) NPO problems, and allow enforcing a new type of constraints: namely, restricting the optimization over states to the set of common ground states of an arbitrary number of operators. Operator optimality conditions are the non-commutative analogs of the Karush--Kuhn--Tucker (KKT) conditions, which are known to hold in many classical optimization problems. In this regard, we prove that a weak form of non-commutative operator optimality holds for all Archimedean NPO problems; stronger versions require the problem constraints to satisfy some qualification criterion, just like in the classical case. We test the power of the new optimality conditions by computing local properties of ground states of many-body spin systems and the maximum quantum violation of Bell inequalities.