Facial Reduction and Partial Polyhedrality

TL;DR

This paper introduces FRA-Poly, a facial reduction algorithm tailored for conic linear programs with polyhedral faces, significantly reducing iterations and improving bounds in specific cases like the doubly nonnegative cone.

Contribution

FRA-Poly is a novel facial reduction method that separately handles polyhedral constraints, leading to fewer iterations and better bounds than classical FRA in certain cones.

Findings

FRA-Poly reduces iteration count in polyhedral cases.

Provides improved worst-case bounds for doubly nonnegative cone.

Proves variants of Gordan-Stiemke's Theorem and separation theorems.

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 idea enables us to reduce 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 $\mathcal{O}(n^2)$. 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.