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Reliably Learn to Trim Multiparametric Quadratic Programs via Constraint Removal

Zhinan Hou, Keyou You

TL;DR

The paper tackles the challenge of solving mp-QPs with many redundant linear inequalities on limited hardware by learning from previously solved instances to safely remove constraints without changing the optimum. It introduces a Lipschitz-based framework to certify that a trimmed mp-QP preserves the original solution, and provides explicit bounds and LICQ-based guarantees. The approach extends to MPC, showing that adaptive online trimming can reduce the online problem to a constraint-free form after a finite time, with additional gains from incorporating offline solved mp-QPs via Voronoi-based selection. Empirical results on a mass-spring system demonstrate substantial reductions in inequality constraints and computation time, highlighting a practical path toward efficient embedded MPC that leverages both offline and online data.

Abstract

In a wide range of applications, we are required to rapidly solve a sequence of convex multiparametric quadratic programs (mp-QPs) on resource-limited hardwares. This is a nontrivial task and has been an active topic for decades in control and optimization communities. Observe that the main computational cost of existing solution algorithms lies in addressing many linear inequality constraints, though their majority are redundant and removing them will not change the optimal solution. This work learns from the results of previously solved mp-QP(s), based on which we propose novel methods to reliably trim (unsolved) mp-QPs via constraint removal, and the trimmed mp-QPs can be much cheaper to solve. Then, we extend to trim mp-QPs of model predictive control (MPC) whose parameter vectors are sampled from linear systems. Importantly, both online and offline solved mp-QPs can be utilized to adaptively trim mp-QPs in the closed-loop system. We show that the number of linear inequalities in the trimmed mp-QP of MPC decreases to zero in a finite timestep, which also can be reduced by increasing offline computation. Finally, simulations are performed to demonstrate the efficiency of our trimming method in removing redundant constraints.

Reliably Learn to Trim Multiparametric Quadratic Programs via Constraint Removal

TL;DR

The paper tackles the challenge of solving mp-QPs with many redundant linear inequalities on limited hardware by learning from previously solved instances to safely remove constraints without changing the optimum. It introduces a Lipschitz-based framework to certify that a trimmed mp-QP preserves the original solution, and provides explicit bounds and LICQ-based guarantees. The approach extends to MPC, showing that adaptive online trimming can reduce the online problem to a constraint-free form after a finite time, with additional gains from incorporating offline solved mp-QPs via Voronoi-based selection. Empirical results on a mass-spring system demonstrate substantial reductions in inequality constraints and computation time, highlighting a practical path toward efficient embedded MPC that leverages both offline and online data.

Abstract

In a wide range of applications, we are required to rapidly solve a sequence of convex multiparametric quadratic programs (mp-QPs) on resource-limited hardwares. This is a nontrivial task and has been an active topic for decades in control and optimization communities. Observe that the main computational cost of existing solution algorithms lies in addressing many linear inequality constraints, though their majority are redundant and removing them will not change the optimal solution. This work learns from the results of previously solved mp-QP(s), based on which we propose novel methods to reliably trim (unsolved) mp-QPs via constraint removal, and the trimmed mp-QPs can be much cheaper to solve. Then, we extend to trim mp-QPs of model predictive control (MPC) whose parameter vectors are sampled from linear systems. Importantly, both online and offline solved mp-QPs can be utilized to adaptively trim mp-QPs in the closed-loop system. We show that the number of linear inequalities in the trimmed mp-QP of MPC decreases to zero in a finite timestep, which also can be reduced by increasing offline computation. Finally, simulations are performed to demonstrate the efficiency of our trimming method in removing redundant constraints.

Paper Structure

This paper contains 24 sections, 15 theorems, 87 equations, 6 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation and our objective
  3. The convex multiparametric quadratic program (mp-QP)
  4. The objective of this work
  5. Reliably learn to trim mp-QPs via safe constraint removal
  6. The trimmed mp-QP
  7. On the zero optimality gap of the trimmed mp-QP
  8. Novel methods to reliably trim mp-QPs from a solved one
  9. Learn to trim from multiple solved mp-QPs
  10. Performance Analysis of Algorithm \ref{['alg:brief']}
  11. Formulas to compute global Lipschitz constant
  12. Performance evaluation of Algorithm \ref{['alg:brief']}
  13. Application to the linear quadratic MPC
  14. The linear quadratic MPC
  15. Adaptively trim mp-QPs of MPC in the closed-loop system
  16. ...and 9 more sections

Key Result

Lemma 1

(nouwens2023constraint) Let the set-valued mapping $\mathbb{C}: \mathcal{R}^{n_x} \rightrightarrows \mathbb{N}_{[1,n_c]}$ denote the index set of removed inequality constraints, i.e., $\mathbb{C}(x) = \mathbb{N}_{[1,n_c]} - \mathbb{I}(x)$. If there exists a mapping $\mathcal{M}: \mathcal{R}^{n_x} \r the trimmed mp-QP in 2_6 does not change the optimal solution, i.e., $z^*(x, \mathbb{I}(x)) = z^*(x

Figures (6)

  • Figure 1: An illustration of Lemma \ref{['lem:outer']}. The black dashed lines denote isolines of the cost function. The orange region is $\mathcal{M}(x)$, and the green solid and dashed lines represent the planes that are specified by the linear inequalities of $\mathbb{I}(x)$ and $\mathbb{C}(x)$, respectively. The left half-space of the green dashed line is the set $\mathcal{Z}(x,\mathbb{C}(x))$.
  • Figure 2: A geometrical illustration of $\mathcal{B}(z^*(\widehat{x}), \kappa \Vert x - \widehat{x} \Vert) \not\subseteq \mathcal{Z}_j(x)$.
  • Figure 3: Voronoi diagrams. The rectangular region represents the feasible set $\mathcal{X}$, which is partitioned into five Voronoi cells using \ref{['Voronoi']}.
  • Figure 4: An oscillating mass system. The arrow denotes the direction of the force that is exerted to masses.
  • Figure 5: The top subfigure shows the percentage of computation time with respect to the standard MPC and the bottom subfigure shows the percentage of inequality constraints remaining in the trimmed MPC.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Lemma 1
  • Definition 1: Global Lipschitz constant (GLC)
  • Remark 1
  • Lemma 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Example 1
  • Theorem 2
  • ...and 14 more