Extracting eigenvalue optimisers in optimization of non commutative polynomials can be done efficiently by Gelfand Naimark-Segal (GNS) construction if the dual (moment) problem has flat optimum solution. However, in numerical computations the flatness is always subject to rounding threshold, i.e., often we can find only approximately flat dual solutions. In this talk we present how to apply GNS construction to approximately flat data and present sensitivity analysis results. We show that if the optimum of the dual problem is close to a flat solution then it yields a solution on the primal side that is close to an optimum solution. The distance to the optimum solution on the primal side can be expressed by the distance to a flat solution on the dual side. Similarly we can express for constraint optimization how close is the solution given by GNS construction to primal feasible and/or optimum solution in terms of the distance of the dual optimum to flatness. With extensive numerical evaluations we show that the established relations are usually very tight when we deal with random non-commutative polynomials. The focus will be on the (constrained) eigenvalue optimization for noncommutative polynomials, but we will also explain how the main results pertain to commutative and tracial optimization