An Improved Primal-Dual Interior Point Solver for Model Predictive Control

TL;DR

This paper introduces an enhanced primal-dual interior-point solver for quadratic programming in MPC, replacing the damped Newton phase with a dual fast gradient method to improve efficiency and convergence.

Contribution

The paper presents a novel primal-dual interior-point solver that integrates a dual fast gradient scheme, reducing computational effort and improving convergence in MPC applications.

Findings

Quadratic convergence to a suboptimal MPC solution.

Reduced computational effort by eliminating backtracking line search.

Effective on unstable systems and aerospace applications.

Abstract

We propose a primal-dual interior-point (PDIP) method for solving quadratic programming problems with linear inequality constraints that typically arise form MPC applications. We show that the solver converges (locally) quadratically to a suboptimal solution of the MPC problem. PDIP solvers rely on two phases: the damped and the pure Newton phases. Compared to state-of-the-art PDIP methods, our solver replaces the initial damped Newton phase (usually used to compute a medium-accuracy solution) with a dual solver based on Nesterov's fast gradient scheme (DFG) that converges with a sublinear convergence rate of order O(1/k^2) to a medium-accuracy solution. The switching strategy to the pure Newton phase, compared to the state of the art, is computed in the dual space to exploit the dual information provided by the DFG in the first phase. Removing the damped Newton phase has the additional advantage that our solver saves the computational effort required by backtracking line search. The effectiveness of the proposed solver is demonstrated on a 2-dimensional discrete-time unstable system and on an aerospace application.