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

Abstract

When solving a quadratic program (QP), one can improve the numerical stability of any QP solver by performing proximal-point outer iterations, resulting in solving a sequence of better conditioned QPs. In this paper we present a method which, for a given multi-parametric quadratic program (mpQP) and any polyhedral set of parameters, determines which sequences of QPs will have to be solved when using outer proximal-point iterations. By knowing this sequence, bounds on the worst-case complexity of the method can be obtained, which is of importance in, for example, real-time model predictive control (MPC) applications. Moreover, we combine the proposed method with previous work on complexity certification for active-set methods to obtain a more detailed certification of the proximal-point method's complexity, namely the total number of inner iterations.

A Linear Programming Method Based on Proximal-Point Iterations With Applications to Multi-Parametric Programming

“…Remark 2 (Complexity certification): If Algorithm 1 is used to solve LPs in the context of linear MPC, the certification method presented in [21] can be used to determine a worstcase complexity bound for Algorithm 1 for a given MPC problem.…”