The corner point method evaluates the objective function only at the vertices of the feasible region, as the optimal solution in linear programming always lies at one such extreme point. If the region is unbounded, you must check if the objective function is being maximized or minimized in the direction of unboundedness to determine if a finite optimum exists. In my own research on optimizing supply chains for tribal farmers, I actually had to discard the corner point method entirely for a critical sub-problem because the feasible region, while bounded, had far too many vertices; I instead used an interior-point method, which was a small triumph in simplifying my model, much like getting a perfect batch of achar on the first try.
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