I have a linear program: minimise $f^T x$ with equality and inequality constraints.
This does not have a unique solution, so I would like to find the solution of this that also minimises $g^T x$.
One approach to doing this is to solve a second linear program with the additional constraint $f^T x = f_{min}$.
In my case, it seems that finding this second minimum takes significantly longer than finding the first.
Is there any way of expressing this process as a single linear program? Or any other better way of tackling the problem?