I'm approaching some continuous optimization problems by considering discrete approximations of them at different resolutions. Those discrete approximations can be solved with linear programming solvers, like GLPK.
After having solved a discrete approximation at resolution $h_0$, I can produce a reasonably good feasible solution $x_h$ for any discrete approximation of the problem at a higher resolution $h$. BUT, that reasonably good feasible solution $x_h$ is in general not a corner(i.e. basic) solution.
Given that solvers like GLPK can do a warm start only at a basic solution, how can I come up with a basic feasible solution near $x_h$ to warm start the solver with? Are there any open source, or at least free, solvers that can be warm started with feasible non-basic solutions?