Exact Complexity Certification of a Nonnegative Least-Squares Method for Quadratic Programming

Abstract

In this paper we propose a method to exactly certify the complexity of an active-set method which is based on reformulating strictly convex quadratic programs to nonnegative least-squares problems. The exact complexity of the method is determined by proving the correspondence between the method and a standard primal active-set method for quadratic programming applied to the dual of the quadratic program to be solved. Once this correspondence has been established, a complexity certification method which has already been established for the primal active-set method is used to also certify the complexity of the nonnegative least-squares method. The usefulness of the proposed method is illustrated on a multi-parametric quadratic program originating from model predictive control of an inverted pendulum.

A Dual Active-Set Solver for Embedded Quadratic Programming Using Recursive LDL$^{T}$ Updates

“…In particular, the active-set method in [1] (QPNNLS), which is based on reformulating the QP as a nonnegative leastsquares (NNLS) problem, is simple to implement and has proven to be efficient for solving small to medium size QP problems. Furthermore, its reliability has been improved greatly in [12] where outer proximal-point iterations are used to improve its numerical stability, and in [13], where QPNNLS is shown to be closely related to a primal active-set QP method applied to the dual problem, allowing the complexity certification method in [14] to be used to determine the exact computational complexity of QPNNLS.…”

“…In this technical note we use insights from [13] to propose a dual active-set method for quadratic programming which retains the favorable properties of QPNNLS (efficiency and simplicity) by making recursive updates to an LDL T factorization. In addition to retaining favorable properties, we show that operating directly on the dual QP instead of the NNLS reformulation used in [1] yields additional improvements: (i) Direct reusability of matrix factors when the linear term in the objective function and the constant term in the constraints change, which is relevant for MPC and when the activeset method is combined with outer proximal-point iterations.…”

Complexity Certification of Proximal-Point Methods for Numerically Stable Quadratic Programming

“…Another recent development for improving the reliability of active-set QP methods are complexity certification methods which, given an mpQP, determines exactly which sequence of active-set changes will occur during solution for any polyhedral set of parameters. In particular, complexity certification methods for the active-set QP methods in [1]- [4] have been presented in [7]- [10], respectively. Furthermore, a general complexity certification framework which encapsulates [7]- [9] has been presented in [11].…”

“…In particular, complexity certification methods for the active-set QP methods in [1]- [4] have been presented in [7]- [10], respectively. Furthermore, a general complexity certification framework which encapsulates [7]- [9] has been presented in [11].…”

Complexity Certification of Proximal-Point Methods for Numerically Stable Quadratic Programming

“…Another recent development for improving the reliability of active-set QP methods are complexity certification methods which, given an mpQP, determines exactly which sequence of active-set changes will occur during solution for any polyhedral set of parameters. In particular, complexity certification methods for the active-set QP methods in [1]- [4] have been presented in [7]- [10], respectively. Furthermore, a general complexity certification framework which encapsulates [7]- [9] has been presented in [11].…”