I'm an engineer working in optimization. As you know, finite, unbounded, real polyhedra are the natural ambient for linear optimization, and much theory is devoted to finding the best solution fast.
Nevertheless, I haven't been able to find a theoretical framework which allows me to "visualize" the structure of large polyhedra.
I tried to look at the face lattice, for example. Vertices can be easily understood as kind of equivalent solutions, but, for instance, I can't grasp the meaning of two vertex not being part of any 1-dimensional face of the polyhedra, but then appearing later together in a higher dimensional face.
Are there any framework for extracting some kind of structural information out of arbitrary polyhedra ? If you consider the face lattice to be one of them: which properties of it (ie chains, antichains, etc) ?
I'm specifically interested in rational cones.