Irreducible factorization of pointed convex cones #41316
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Hauser and Güler's paper "Self-Scaled Barrier Functions on Symmetric Cones and Their Classification" is one of the many places where it is proved that pointed convex cones can be decomposed into irreducible factors. This one is nice because the proof is simple (linear algebra in R^n), and it provides uniqueness when the cone is solid.
The paper "Computing symmetry groups of polyhedra" by Bremner et al. sketches an algorithm for computing the irreducible factors of a polyhedral convex cone.
This was effectively suggested in PR 40367 to compute linear isomorphisms of cones. To find isomorphisms, we would like to decompose the cones into irreducible factors, and then look for isomorphisms between factors of the appropriate size. Since the isomorphism algorithm is combinatorial, it is better to work with lots of small cones than it is to work with one big cone. In any case, this is of independent interest and is not too hard to implement. Thanks to Dmitrii Pasechnik for the suggestion (and the algorithm).
Add a new method for people who don't care about the details of the factorization and who only need to know whether or not their cone is reducible.
Permutation invariant proper polyhedral cones and their Lyapunov rank, by Jeong and Gowda. There's a result in here that provides an example for the new is_reducible() method of polyhedral convex cones.
Combine two results from the literature to make a reducibility example out of cones.rearrangement().
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Every pointed convex cone can be factored in to a direct sum of irreducible (i.e. cannot be factored further) parts. When the cones are polyhedral (as they are in sage), this factorization can be computed. This PR adds two methods to cone objects:
irreducible_factors(), to return the factorizationis_reducible(), for when you only care about the number of irreducible componentsThere are many places in optimization, game theory, and even graph theory where some cone being irreducible is important. This was indirectly suggested by @dimpase in #40367 where factoring can decrease the time it takes to compute linear isomorphisms.