My understanding of strong duality (for linear programming) is that if the dual form is optimal, then the primal is also optimal, and vice versa. Therefore, we only need to optimize one problem (either the primal or the dual), since the optimal value of those two problems are the same.
However, in this case, how can we know we have achieved the optimal solution for our single problem? A naive check (using weak duality) is that we can optimize the opposite problem and see if it reaches a value that is equal to the objective function value of our problem, but this seems way too excessive since we need to optimize both problems. In other words, is there another check (without using the opposite problem) that could guarantee that our solution is optimal and we can stop the optimization?
If my understanding of strong duality is inaccurate in any place, please feel free to correct me. Thank you in advance.