Active-set prediction for interior point methods using controlled perturbations

TL;DR

This paper introduces a method using controlled perturbations in linear programming to improve active-set prediction for interior point methods, enabling earlier and more accurate identification of the optimal active set.

Contribution

It proposes a novel approach of perturbing constraints to enhance active-set prediction accuracy and timing in interior point methods for linear programming.

Findings

Perturbations enlarge the feasible set and help predict the active set.

The primal-dual path-following algorithm predicts the active set accurately when the duality gap is not too small.

Numerical experiments show promising results comparing perturbed and unperturbed formulations.

Abstract

We propose the use of controlled perturbations to address the challenging question of optimal active-set prediction for interior point methods. Namely, in the context of linear programming, we consider perturbing the inequality constraints/bounds so as to enlarge the feasible set. We show that if the perturbations are chosen appropriately, the solution of the original problem lies on or close to the central path of the perturbed problem. We also find that a primal-dual path-following algorithm applied to the perturbed problem is able to accurately predict the optimal active set of the original problem when the duality gap for the perturbed problem is not too small; furthermore, depending on problem conditioning, this prediction can happen sooner than predicting the active set for the perturbed problem or when the original one is solved. Encouraging preliminary numerical experience is reported when comparing activity prediction for the perturbed and unperturbed problem formulations.