Algorithmic Complexity of Statistical Estimators
The bound $O(Q_n(θ) \cdot S(p))$ represents only the cost of evaluating $Q_n$ once per step in the "numerical optimization method"; you ignore all other cost that incurs.

Without looking at the whole algorithm, little more can be done.

Note: I very deliberately replaced $S$ with $S(p)$. That is because there is no reason, per se, to believe that $S$ is a constant. You need to be more careful about setting up your cost functions; our reference question may be helpful.