the convergence of the iterative algorithm has a major problem

In order to solve an optimization problem, I divided the main problem into two sub-problems. The two sub-problems require to be solved iteratively until the algorithm converges.

I use the bi-section search followed by the (LLL) lattice-reduction algorithm to solve the first sub-problem. In the second sub-problem, I used the barrier method to solve it.

However, the solution of the first sub-problem depends on a sub-optimal, say if not heuristic, algorithm, then the alternating algorithm often easily terminates due to no or even adverse progress. How do I solve this kind of problem?

I suggest running the iterative algorithm over many realizations and taking the average over these realizations at the end. But, I am not sure if this solution is valid or not.