A New Active Set Scheme for Quadratic Programing

TL;DR

This paper introduces a novel active set algorithm for convex quadratic programming that leverages matrix properties and singular value decomposition to improve computational efficiency, especially in problems with many constraints.

Contribution

The paper proposes new active set schemes based on matrix properties and null space computation, enhancing solution speed for large-scale convex quadratic problems.

Findings

Reduced computation time with larger problems.

Second scheme outperforms original in high-constraint scenarios.

Validated over 2000 tests on random and standard problems.

Abstract

We are faced with convex quadratic programing in many contexts related to control theory, economy and robotics. In this paper, we introduce a new active set algorithm for solving such problems and analyze its possible advantages. The novelty of the proposed scheme is in the way of solving the KKT system based on matrix properties. More precisely, we combine the two KKT equations to reduce the order and substitute it with a null space computation. The null space is in hand by use of the singular values decomposition. In problems with high number of independent constraints, we proposed another scheme. This also aims to solve the KKT system based on matrix properties. We implement both algorithms and test them over both randomly generated problems and standard problems mentioned in CUTEst. In general, more than 2000 tests with great variety are generated and computing times and accuracies are reported. The proposed schemes for solving convex quadratic problems are members of active set family. Because of using matrix properties, it reduces computing time and as larger as the problem is, the improvement shows to be more remarkable. The first strategy performs the original active set when the number of constraints is low while the second outperforms the original algorithm when there exists a lot of independent constraints.