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A High-Performant Multi-Parametric Quadratic Programming Solver

Daniel Arnström, Daniel Axehill

TL;DR

The paper tackles the efficient computation of explicit solutions to multi-parametric quadratic programs (mpQP) that underpin explicit MPC. It recasts mpQP as an equivalent multi-parametric least-distance problem (mpLDP) and employs KKT conditions with active sets to describe piecewise-affine solutions over polyhedral critical regions. The core contribution is a combinatorial connected-graph algorithm that traverses combinatorially adjacent active sets, proving that LICQ-satisfying active sets form a connected graph and avoiding heavy geometric facet computations. Empirical results show the approach yields around a two-orders-of-magnitude speedup over leading mpQP solvers like MPT and POP, highlighting significant practical impact for real-time explicit MPC and related control applications. The work also clarifies degeneracy handling and provides a path toward parallelized implementations.

Abstract

We propose a combinatorial method for computing explicit solutions to multi-parametric quadratic programs, which can be used to compute explicit control laws for linear model predictive control. In contrast to classical methods, which are based on geometrical adjacency, the proposed method is based on combinatorial adjacency. After introducing the notion of combinatorial adjacency, we show that the explicit solution forms a connected graph in terms of it. We then leverage this connectedness to propose an algorithm that computes the explicit solution. The purely combinatorial nature of the algorithm leads to computational advantages since it enables demanding geometrical operations (such as computing facets of polytopes) to be avoided. Compared with classical combinatorial methods, the proposed method requires fewer combinations to be considered by exploiting combinatorial connectedness. We show that an implementation of the proposed method can yield a speedup of about two orders of magnitude compared with state-of-the-art software packages such as MPT and POP.

A High-Performant Multi-Parametric Quadratic Programming Solver

TL;DR

The paper tackles the efficient computation of explicit solutions to multi-parametric quadratic programs (mpQP) that underpin explicit MPC. It recasts mpQP as an equivalent multi-parametric least-distance problem (mpLDP) and employs KKT conditions with active sets to describe piecewise-affine solutions over polyhedral critical regions. The core contribution is a combinatorial connected-graph algorithm that traverses combinatorially adjacent active sets, proving that LICQ-satisfying active sets form a connected graph and avoiding heavy geometric facet computations. Empirical results show the approach yields around a two-orders-of-magnitude speedup over leading mpQP solvers like MPT and POP, highlighting significant practical impact for real-time explicit MPC and related control applications. The work also clarifies degeneracy handling and provides a path toward parallelized implementations.

Abstract

We propose a combinatorial method for computing explicit solutions to multi-parametric quadratic programs, which can be used to compute explicit control laws for linear model predictive control. In contrast to classical methods, which are based on geometrical adjacency, the proposed method is based on combinatorial adjacency. After introducing the notion of combinatorial adjacency, we show that the explicit solution forms a connected graph in terms of it. We then leverage this connectedness to propose an algorithm that computes the explicit solution. The purely combinatorial nature of the algorithm leads to computational advantages since it enables demanding geometrical operations (such as computing facets of polytopes) to be avoided. Compared with classical combinatorial methods, the proposed method requires fewer combinations to be considered by exploiting combinatorial connectedness. We show that an implementation of the proposed method can yield a speedup of about two orders of magnitude compared with state-of-the-art software packages such as MPT and POP.
Paper Structure (12 sections, 6 theorems, 15 equations, 1 figure, 1 algorithm)

This paper contains 12 sections, 6 theorems, 15 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivalent least-distance problem
  4. The explicit solution
  5. Existing methods to compute the explicit solution
  6. A combinatorial connected-graph algorithm
  7. Geometrical and combinatorial adjacency
  8. A combinatorial algorithm
  9. Comparison with similar methods
  10. Numerical Experiments
  11. Conclusion
  12. Proof Lemma \ref{['lem:geocomb']}

Key Result

Lemma 1

The offset $d: \mathbb{R}^p \to \mathbb{R}^m$ is an affine function of $\theta$.

Figures (1)

  • Figure 1: Time taken for MPTherceg2013mpt, POPoberdieck2017explicit, and a Julia implementation of Algorithm \ref{['alg:updown']} ("Proposed"), to compute the explicit solution for the benchmark mpQP problems from the POP toolbox oberdieck2016pop.

Theorems & Definitions (26)

  • Lemma 1: Affine offset
  • proof
  • Remark 1: Relating mpLDP and mpQP
  • Definition 1: Active set
  • Definition 2: Critical region
  • Definition 3: LICQ
  • Remark 2: Explicit solution to mpQP
  • Definition 4: Geometrical adjacency
  • Definition 5: Combinatorial adjacency
  • Definition 6: Valid combinatorial sequence
  • ...and 16 more