Homogeneous Formulation of Convex Quadratic Programs for Infeasibility Detection

TL;DR

This paper introduces a homogeneous formulation of convex quadratic programs that guarantees feasibility, simplifies infeasibility detection, and integrates with existing algorithms, enhancing robustness and computational efficiency.

Contribution

The paper proposes a novel homogeneous QP formulation that ensures feasibility and enables straightforward infeasibility detection, compatible with existing algorithms.

Findings

The homogeneous QP always remains feasible.

Infeasibility detection is simplified through the new formulation.

The proposed Infeasible Interior Point Method has polynomial complexity.

Abstract

Convex Quadratic Programs (QPs) have come to play a central role in the computation of control action for constrained dynamical systems. In this paper, we present a novel Homogeneous QP (HQP) formulation which is obtained by embedding the original QP in a larger space. The key properties of the HQP are: (i) is always feasible, (ii) an optimal solution to QP can be readily obtained from a solution to HQP, and (iii) infeasibility of QP corresponds to a particular solution of HQP. An immediate consequence is that all the existing algorithms for QP are now also capable of robustly detecting infeasibility. In particular, we present an Infeasible Interior Point Method (IIPM) for the HQP and show polynomial iteration complexity when applied to HQP. A key distinction with prior IPM approaches is that we do not need to solve second-order cone programs. Numerical experiments on the formulation are provided using existing codes.