Facial Reduction and Partial Polyhedrality

We present FRA-Poly, a facial reduction algorithm (FRA) for conic linear programs that is sensitive to the presence of polyhedral faces in the cone. The main goals of FRA and FRA-Poly are the same, i.e., finding the minimal face containing the feasible region and detecting infeasibility, but FRA-Poly treats polyhedral constraints separately. This reduces the number of iterations drastically when there are many linear inequality constraints. The worst-case number of iterations for FRA-Poly is written in terms of a "distance to polyhedrality" quantity and provides better bounds than FRA under mild conditions. In particular, in the case of the doubly nonnegative cone, FRA-Poly gives a worst-case bound of n whereas the classical FRA bound is O(n(2)). Of possible independent interest, we prove a variant of the Gordan-Stiemke theorem and a proper separation theorem which takes into account partial polyhedrality. We provide a discussion on the optimal facial reduction strategy and an instance that forces FRAs to perform many steps. We also present a few applications. In particular, we will use FRA-Poly to improve the bounds recently obtained by Liu and Pataki on the dimension of certain affine subspaces which appear in weakly infeasible problems.