When maximizing an objective function with the simplex algorithm, if there exist a positive reduced cost with all negative entries in the column, we then know that the solution is unbounded. The question is, is there a way to sniff out possible unboundedness before even starting the simplex algorithm?
For example,
- Producing X requires 3 of resource A, and creates 3 of resource B as a by-product.
- Producing Y requires 3 if resource B, and creates 4 if resource A as a by-product.
This is not from a real scenario, merely an example of conditions that will allow us to create infinite amount of product X, and thus there is no optimal solution, the maximum objective value is unbounded.
One possibility is to look at the dual problem. If the dual is infeasible, and you know the primal problem is feasible, then the primal problem must have an unbounded optimum. However, this depends on whether it is easy to see the dual is infeasible. In general, it is as difficult detecting infeasibility as solving a linear program. So I guess, the answer to your question in general is no, unless the problem has some special structure.