Facial Reduction and Partial Polyhedrality

Abstract

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 the 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 is O(n 2 ). Of possible independent interest, we prove a variant of Gordan-Stiemke's Theorem and a proper separation theorem that 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.

Amenable cones: error bounds without constraint qualifications

“…The PPS condition reflects the fact that we only care about having a relative interior point with respect the part of the cone that we know that is not polyhedral. When a conic linear program satisfies the PPS condition, we get the same consequences of the usual Slater's condition: zero duality gap and, when the optimal value is finite, the dual problem is attained (e.g., Proposition 23 in [30]).…”

“…Sometimes it is enough to find a face that satisfies a less strict constraint qualification. In particular, the FRA-Poly algorithm in [30] is divided in two phases. In the first phase, a face satisfying the PPS condition is found and in the second phase, F min is computed.…”

“…In the first phase, a face satisfying the PPS condition is found and in the second phase, F min is computed. In many cases of interest, this two-phase strategy leads to better bounds on the singularity degree than the classical facial reduction algorithm, see for instance, Table 1 in [30]. We will recall here a few definitions and results from [30].…”

“…See Example 1 in [30] for the values of ℓ poly (K) for some common cones. In particular, if K is polyhedral, we have ℓ poly (K) = 0.…”

“…This is where we will use facial reduction (Section 2.4). If (K, L, a) is feasible, but the PPS condition is not satisfied, then there exists z 1 ∈ K * ∩ L ∩ {a} ⊥ with z 1 ∈ K ⊥ , e.g., Theorem 4 in [30]. In particular, F 1 := K ∩ {z 1 } ⊥ is a proper face of K that contains the feasible region of (K, L, a).…”