[Paper Review] An algorithm to compute the Hoffman constant of a system of linear constraints

Javier Peña, Juan C. Vera+1 more|arXiv (Cornell University)|Apr 23, 2018

Advanced Optimization Algorithms Research32 references3 citations

TL;DR

This paper presents a combinatorial algorithm to compute the Hoffman constant for systems of linear constraints, including those with easy-to-satisfy constraints such as box bounds. The method leverages a novel characterization of the Hoffman constant as the maximum over a finite, tractable collection of subproblems, enabling exact computation via convex optimization on well-posed subsystems.

ABSTRACT

We propose a combinatorial algorithm to compute the Hoffman constant of a system of linear equations and inequalities. The algorithm is based on a characterization of the Hoffman constant as the largest of a finite canonical collection of easy-to-compute Hoffman constants. Our algorithm and characterization extend to the more general context where some of the constraints are easy to satisfy as in the case of box constraints. We highlight some natural connections between our characterizations of the Hoffman constant and Renegar's distance to ill-posedness for systems of linear constraints.

Motivation & Objective

• To develop an efficient algorithm for computing the exact Hoffman constant of a system of linear inequalities and equations.
• To extend the computation of the Hoffman constant to systems that include easy-to-satisfy constraints, such as bound constraints.
• To provide a characterization of the Hoffman constant based on surjective submapping properties, enabling tractable computation.
• To establish connections between the Hoffman constant and Renegar’s distance to ill-posedness for linear systems.
• To support modern optimization algorithms by providing computable error bounds for feasibility and optimality.

Proposed method

• The algorithm is based on a characterization of the Hoffman constant as the maximum over all subsets $ J otin ext{S}(A) $ where the submapping $ x \mapsto A_J x + \mathbb{R}^J_+ $ is surjective.
• It computes $ H(A) = \max_{J \in \mathcal{S}(A)} \frac{1}{\min_{v \in \mathbb{R}^J_+, \|v\|^* = 1} \|A_J^T v\|^*} $, reducing the search to a finite, manageable set of subsets.
• The method identifies a smaller canonical collection $ \mathcal{F} \subseteq \mathcal{S}(A) $ that suffices to compute the maximum, avoiding brute-force enumeration.
• It uses convex optimization to compute the inner minimization $ \min_{v \in \mathbb{R}^J_+, \|v\|^* = 1} \|A_J^T v\|^* $ for each candidate $ J $.
• The framework extends to systems with both equations and inequalities, and to cases with constraints that are trivially satisfiable (e.g., box bounds).
• Theoretical foundations are grounded in poyhedral sublinear mappings and facial reduction, with the Hoffman constant linked to the inverse norm of certain linear mappings.

Experimental results

Research questions

• RQ1 How can the Hoffman constant be computed exactly for general systems of linear constraints, including those with easy-to-satisfy constraints?
• RQ2 What structural properties of the constraint matrix $ A $ enable efficient computation of the Hoffman constant?
• RQ3 How does the Hoffman constant relate to Renegar’s distance to ill-posedness in linear systems?
• RQ4 Can a finite, computationally tractable set of subsystems capture the worst-case error bound in Hoffman-type inequalities?
• RQ5 What is the role of well-posedness of subsystems in characterizing the global Hoffman constant?

Key findings

• The Hoffman constant $ H(A) $ is exactly characterized as the maximum over a finite set of easily computable subproblems, each corresponding to a well-posed subsystem.
• The algorithm computes $ H(A) $ by evaluating $ \frac{1}{\min_{v \in \mathbb{R}^J_+, \|v\|^* = 1} \|A_J^T v\|^*} $ over a canonical subset $ \mathcal{F} \subseteq \mathcal{S}(A) $, avoiding exponential enumeration.
• The method generalizes to systems with both equations and inequalities, and to cases with box constraints, where such constraints are treated as trivially satisfiable.
• The facial distance or pyramidal width in recent optimization methods is shown to be a special case of this generalized Hoffman constant.
• The characterization establishes a direct link between the Hoffman constant and Renegar’s distance to ill-posedness: $ 1/H(A) $ equals the smallest perturbation that makes $ Ax < 0 $ infeasible.
• The framework provides a unified approach to error bounds via poyhedral sublinear mappings, enabling systematic derivation of Hoffman-type constants.