Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Parallel Vector-form $LDL^\top$ Decomposition for Accelerating Execution-time-certified $\ell_1$-penalty Soft-constrained MPC

Liang Wu, Liwei Zhou, Richard D. Braatz

TL;DR

The paper tackles the challenge of providing execution-time certificates for real-time MPC while also ensuring feasibility under disturbances. It demonstrates that an $\,\ell_1$-penalty soft-constrained MPC can be transformed into a Box-QP and solved with a prior Box-QP certificate methodology, while introducing a novel parallel vector-form $LDL^\top$ decomposition to accelerate Newton steps. This combination yields large computational speedups (up to $O(1000\times)$) and enables performance comparable to IPOPT and OSQP, with an implementation available publicly. The results include theoretical guarantees, a scalable algorithmic framework, and compelling numerical experiments (including an AFTI-F16 MPC example with infeasibility) that highlight practical impact for real-time constrained control.

Abstract

Handling possible infeasibility and providing an execution time certificate are two pressing requirements of real-time Model Predictive Control (MPC). To meet these two requirements simultaneously, this paper proposes an $\ell_1$-penalty soft-constrained MPC formulation that is globally feasible and solvable with an execution time certificate using our proposed algorithm. This paper proves for the first time that $\ell_1$-penalty soft-constrained MPC problems can be equivalently transformed into a box-constrained quadratic programming (Box-QP) and then our previous execution-time-certified algorithm \cite{wu2023direct} (only limited to Box-QP) can be applied. However, our previous Box-QP algorithm \cite{wu2023direct}, which provides a theoretical execution-time certificate, is conservative in its iteration analysis, thus sacrificing computation efficiency. To this end, this paper proposes a novel $LDL^\top$ decomposition for the first time, to accelerate the computation of Newton step at each iteration. The speedup of our $LDL^\top$ decomposition comes from two-fold: \textit{i)} exploitation of the fact that the number of inequality constraints is generally larger than the number of variables in condensed MPC formulations, \textit{ii)} vectorized and parallel implementation based on based on its vector-wise operations, instead of element-wise operations of previous decomposition methods. Numerical experiments demonstrate great speedups of the proposed $LDL^\top$ decomposition (even up to 1000-fold, compared to the standard Choleksky method), which thus helps our solver achieve comparable computation performance to the state-of-the-art solvers such as IPOPT and OSQP. Code is available at \url{https://github.com/liangwu2019/L1-penalty-QP}.

A Parallel Vector-form $LDL^\top$ Decomposition for Accelerating Execution-time-certified $\ell_1$-penalty Soft-constrained MPC

TL;DR

The paper tackles the challenge of providing execution-time certificates for real-time MPC while also ensuring feasibility under disturbances. It demonstrates that an -penalty soft-constrained MPC can be transformed into a Box-QP and solved with a prior Box-QP certificate methodology, while introducing a novel parallel vector-form decomposition to accelerate Newton steps. This combination yields large computational speedups (up to ) and enables performance comparable to IPOPT and OSQP, with an implementation available publicly. The results include theoretical guarantees, a scalable algorithmic framework, and compelling numerical experiments (including an AFTI-F16 MPC example with infeasibility) that highlight practical impact for real-time constrained control.

Abstract

Handling possible infeasibility and providing an execution time certificate are two pressing requirements of real-time Model Predictive Control (MPC). To meet these two requirements simultaneously, this paper proposes an -penalty soft-constrained MPC formulation that is globally feasible and solvable with an execution time certificate using our proposed algorithm. This paper proves for the first time that -penalty soft-constrained MPC problems can be equivalently transformed into a box-constrained quadratic programming (Box-QP) and then our previous execution-time-certified algorithm \cite{wu2023direct} (only limited to Box-QP) can be applied. However, our previous Box-QP algorithm \cite{wu2023direct}, which provides a theoretical execution-time certificate, is conservative in its iteration analysis, thus sacrificing computation efficiency. To this end, this paper proposes a novel decomposition for the first time, to accelerate the computation of Newton step at each iteration. The speedup of our decomposition comes from two-fold: \textit{i)} exploitation of the fact that the number of inequality constraints is generally larger than the number of variables in condensed MPC formulations, \textit{ii)} vectorized and parallel implementation based on based on its vector-wise operations, instead of element-wise operations of previous decomposition methods. Numerical experiments demonstrate great speedups of the proposed decomposition (even up to 1000-fold, compared to the standard Choleksky method), which thus helps our solver achieve comparable computation performance to the state-of-the-art solvers such as IPOPT and OSQP. Code is available at \url{https://github.com/liangwu2019/L1-penalty-QP}.
Paper Structure (23 sections, 8 theorems, 57 equations, 10 figures, 3 tables, 3 algorithms)

This paper contains 23 sections, 8 theorems, 57 equations, 10 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. Notations
  5. Problem formulation
  6. Transform non-smooth $\ell_1$-penalty QP to Box-QP
  7. Scaling the Box-QP
  8. Execution-time-certified QP algorithm
  9. Strictly feasible initial point
  10. Newton direction
  11. Iteration complexity and algorithm implementation
  12. Parallel Vector-form $LDL^\top$ Decomposition
  13. Decomposition procedure of parallel vector-form $LDL^\top$
  14. Substitution procedure of parallel vector-form $LDL^\top$
  15. Parallel implementation
  16. ...and 8 more sections

Key Result

Lemma 1

Suppose that $Q$ is symmetric positive definite, the $\ell_1$-penalty non-smooth QP problem_QP_to_penaltyQP is globally feasible with a unique optimal solution $y^*$.

Figures (10)

  • Figure : (a) $m=25$
  • Figure : (a)
  • Figure : (a) $m=25$
  • Figure : (b) $m=50$
  • Figure : (c) $m=100$
  • ...and 5 more figures

Theorems & Definitions (15)

  • Remark 1
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Theorem 1
  • proof
  • Corollary 2
  • Lemma 2: See Theorem 2, wu2023direct
  • Lemma 3
  • ...and 5 more