[Paper Review] Facial Reduction and Partial Polyhedrality

Bruno F. Lourenço, Masakazu Muramatsu+1 more|arXiv (Cornell University)|Dec 8, 2015

Advanced Optimization Algorithms Research29 references18 citations

TL;DR

This paper introduces FRA-Poly, a facial reduction algorithm that exploits polyhedral faces in conic optimization to drastically reduce the number of iterations needed for convergence. By treating polyhedral constraints separately and leveraging partial polyhedrality, FRA-Poly achieves a worst-case bound of n iterations for the doubly nonnegative cone Dn—significantly better than the O(n²) bound of classical facial reduction—while also generalizing Gordan-Stiemke's Theorem and providing improved bounds on singularity degrees and weakly infeasible subspaces.

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.

Motivation & Objective

• To reduce the number of facial reduction iterations in conic linear programs by exploiting the presence of polyhedral faces.
• To develop a new facial reduction algorithm, FRA-Poly, that treats polyhedral constraints separately to improve worst-case complexity.
• To provide tighter bounds on the singularity degree of conic programs, especially for the doubly nonnegative cone Dn.
• To generalize classical theorems like Gordan-Stiemke's Theorem to account for partial polyhedrality.
• To improve existing bounds on the dimension of subspaces associated with weakly infeasible problems, as studied by Liu and Pataki.

Proposed method

• FRA-Poly operates in two phases: Phase 1 performs standard facial reduction until a face containing polyhedral blocks is reached; Phase 2 jumps directly to the minimal face in one step.
• The algorithm uses a new quantity, 'distance to polyhedrality' ℓpoly(K), which measures the length of the longest strictly ascending chain of nonempty faces starting from a polyhedral face.
• For a product cone K = K₁ × ⋯ × Kᵣ, FRA-Poly terminates in at most 1 + Σᵢ ℓpoly(Kᵢ) steps, which is strictly smaller than classical FRA when at least two Ki are not subspaces.
• The method relies on a generalized proper separation theorem (Theorem 4) and a variant of Gordan-Stiemke's Theorem (Theorem 5) that accounts for partial polyhedrality.
• It exploits the structure of direct products of cones, where a face Fᵢ = F₁ᵢ × ⋯ × Fᵣᵢ can be reduced to the minimal face if each Fⱼᵢ is either polyhedral or already matches the j-th block of the minimal face.
• The algorithm is applied to the doubly nonnegative cone Dn = Sⁿ₊ ∩ Nⁿ, where it proves that the singularity degree is at most n, improving upon the classical O(n²) bound.

Experimental results

Research questions

• RQ1 Can facial reduction be accelerated by explicitly handling polyhedral constraints separately, rather than treating them uniformly with other cone constraints?
• RQ2 What is the worst-case number of facial reduction steps for conic programs when partial polyhedrality is present, and how does it compare to classical facial reduction?
• RQ3 How does the 'distance to polyhedrality' ℓpoly(K) quantify the complexity of facial reduction in product cones?
• RQ4 Can FRA-Poly provide tighter bounds on the singularity degree of conic programs, particularly for the doubly nonnegative cone Dn?
• RQ5 To what extent can FRA-Poly improve existing bounds on the dimension of subspaces linked to weakly infeasible problems?

Key findings

• FRA-Poly reduces the worst-case number of facial reduction steps for the doubly nonnegative cone Dn from O(n²) to n, a significant improvement over classical facial reduction.
• For a product cone K = K₁ × ⋯ × Kᵣ, FRA-Poly requires at most 1 + Σᵢ ℓpoly(Kᵢ) steps, which is strictly smaller than classical FRA when at least two Ki are not subspaces.
• The paper proves a generalized Gordan-Stiemke's Theorem for cones that are direct products of a closed convex cone and a polyhedral cone, extending classical duality results.
• FRA-Poly enables strong duality to be restored in a single step once a face with polyhedral blocks is reached, even if the face is not the minimal face.
• The singularity degree of any conic linear program over Dn is at most n, which improves upon the classical bound of n(n+1)/2.
• The algorithm improves bounds on the dimension of affine subspaces associated with weakly infeasible problems, as recently studied by Liu and Pataki.