# Dual solution when solving a primal degenerate LP with the interior point algorithm

Say, we're working with an LP that is primal degenerate (optimal solution is at a vertex but with multiple bases) and not dual degenerate (optimal solution is not at a face).

If we were to solve it with a simplex algorithm, we'd get a basic primal solution vector and a basic dual solution vector.

If we were to solve it with an interior point (barrier) algorithm, we'd get a "near-basic" primal solution vector (by near-basic, I mean very close to a vertex but still in the interior of the feasible region). What would the dual solution vector look like in this case? Will it be a "midface" solution?

My intuition is that it should be a midface solution since the dual of this LP will be dual degenerate, and solving this dual LP with the interior point algorithm will give us a midface solution. Is this a good way to think about it? I tested this by solving a two-variable LP example with CPLEX and it seems to confirm this, but I'm not sure if it will follow in higher dimensions.