We did sorting-via-linear-programming as a simple exercise when I was studying maths in the mid 2000s. It was certainly not seen as a research grade problem.

In general if memory serves right, linear programming can solve every problem that's in P with some suitable linear time preprocessing. See https://en.wikipedia.org/wiki/P-complete for some background.

Inner point methods to solve linear programs should then give you the bridge to continuous domains.