Active-set prediction in quadratic programming using interior point methods and controlled perturbations

TL;DR

This paper proposes a method using controlled perturbations in interior point algorithms to improve active-set prediction accuracy in convex quadratic programming, supported by theoretical proofs and preliminary numerical results.

Contribution

It introduces a novel approach of perturbing constraints to better predict active sets in quadratic programming, with theoretical guarantees and initial numerical validation.

Findings

Perturbations can enlarge the feasible set without losing optimal active set information.

The method accurately predicts the active and inactive sets at optimality.

Preliminary numerical experiments show promising results for the proposed approach.

Abstract

In this paper, we extend the idea of using controlled perturbations to enhance the capabilities of active-set prediction for interior point methods for convex Quadratic Programming (QP) problems. Namely, we consider perturbing the inequality constraints (by a small amount) so as to enlarge the feasible set. We show that if the perturbations are chosen judiciously, then there exists a primal-dual pair of points which is close to the optimal solution of the perturbed problems and the corresponding active and inactive sets at this point are the same as the optimal active and inactive sets at an optimal solution of the original QP problems. Additionally, we prove that the optimal tripartition of the original problems can also be predicted by solving the perturbed ones. Furthermore, encouraging preliminary numerical experience is also presented for the QP case.