Linear programming is, of course, nowadays very well understood. We have a lot of work that characterises the structure of feasible solutions, and the structure of optimal solutions. We have the strong duality, poly-time algorithms, etc.
But what is known about minimum maximal solutions of LPs? Or, equivalently, maximum minimal solutions?
(This is not really a research question, but maybe we can have something less technical for the holidays. I'm just being curious, and after some googling I got the feeling that I must be missing the right keywords. It feels like an obvious problem to study, but I only found some sporadic papers that mention the problem.)
To keep things simple, let's focus on packing and covering LPs. In a packing LP we are given a non-negative matrix $A$. A vector $x$ is feasible if $x \ge 0$ and $Ax \le 1$. We say that $x$ is maximal if it is feasible and we can't greedily increase any component. That is, if $y \ge 0$ and $y \ne 0$, then $x + y$ is not feasible. And finally, $x$ is a minimum maximal solution, if it minimises the objective function $\sum_i x_i$ among all maximal solutions.
(You can define a maximum minimal solution of a covering LP in an analogous manner.)
What does the space of minimum maximal solutions look like? How can we find such solutions? How difficult it is to find such solutions? How can we approximate such solutions? Who studies such things, and what is the right term for it?
These questions were originally motivated by edge dominating sets and minimum maximal matchings. It is well known (and fairly easy to see) that a minimum maximal matching is a minimum edge dominating set; conversely, given a minimum edge dominating set, it is easy to construct a minimum maximal matching.
So they are, in essence, the same problem. Both problems are NP-hard and APX-hard. There is a trivial 2-approximation algorithm: any maximal matching.
However, their "natural" LP relaxations look very different. If you take the edge dominating set problem and form a natural LP relaxation, you get a covering LP. However, if you take the problem of finding a minimum maximal matching and try to come up with an LP relaxation, then what do you get? Well, of course fractional matchings are feasible solutions of a packing LP; then maximal fractional matchings are maximal solutions of such LPs, and minimum maximal fractional matchings are therefore minimum maximal solutions of such LPs. :)