Linear Programming (LP) with large number of variables and the similar LP with small number of variables

Question

Assume we are given a Linear Programming (LP) where the number of variables depends on the $N$. This LP can be solved using a LP solver for small N's (e.g., 2,3,4) and has this property that the results for all of these cases follow a similar pattern and similar cost which can be used to guess the solution of the LP for other values of N. For large values of N, assuming that it follows the same pattern, we can guess a solution for the LP (which I believe is actually the optimal solution of LP). My question is that, how can we check that this solution is optimal? Is there any technique?

Answer 1

Yes, all you need to do is check the reduced costs of your solution. For a maximization (minimization) problem, if they are all be negative (positive), the solution is optimal.