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Choosing Augmentation Parameters in OSQP- A New Approach based on Conjugate Directions

Avinash Kumar

TL;DR

The paper targets accelerating OSQP, an ADMM-based solver for convex quadratic programs, by leveraging information about conjugate directions of the coefficient matrix to offline-compute and cache augmentation parameters $\varrho$ and the associated directions. This offline caching enables faster inversion of the critical linear system $P+\sigma I + A^T\varrho A$ during iterations. A conjugate-direction framework (and its CG variant) is used to achieve efficient solves, with an adaptive update rule for $\varrho$ based on primal/dual residuals to further improve convergence. A numerical example demonstrates reduced inversion time $T_{inv}$ and overall time $T_{tot}$, validating the approach for faster real-time QP solving in large-scale settings.

Abstract

This work proposes a new method to select the augmentation parameters in the operator splitting quadratic program (OSQP) algorithm so as to reduce the computation time of overall algorithm. The selection is based upon the information of conjugate directions of the coefficient matrix of a linear system of equations present in the algorithm. This selection makes it possible to cache these conjugate directions, instead of computing them at each iteration, resulting in faster computation of the solution of the linear system thus reducing the overall computation time. This reduction is demonstrated by a numerical example.

Choosing Augmentation Parameters in OSQP- A New Approach based on Conjugate Directions

TL;DR

The paper targets accelerating OSQP, an ADMM-based solver for convex quadratic programs, by leveraging information about conjugate directions of the coefficient matrix to offline-compute and cache augmentation parameters and the associated directions. This offline caching enables faster inversion of the critical linear system during iterations. A conjugate-direction framework (and its CG variant) is used to achieve efficient solves, with an adaptive update rule for based on primal/dual residuals to further improve convergence. A numerical example demonstrates reduced inversion time and overall time , validating the approach for faster real-time QP solving in large-scale settings.

Abstract

This work proposes a new method to select the augmentation parameters in the operator splitting quadratic program (OSQP) algorithm so as to reduce the computation time of overall algorithm. The selection is based upon the information of conjugate directions of the coefficient matrix of a linear system of equations present in the algorithm. This selection makes it possible to cache these conjugate directions, instead of computing them at each iteration, resulting in faster computation of the solution of the linear system thus reducing the overall computation time. This reduction is demonstrated by a numerical example.

Paper Structure

This paper contains 9 sections, 2 theorems, 13 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. OSQP
  4. Conjugate Direction Methods
  5. Main Results
  6. Choosing the set of Conjugate Directions
  7. Adaptive Update of $\varrho$
  8. Numerical Example
  9. Conclusion

Key Result

Theorem 1

nocedal1999numerical For any $\mathbf{\bar{x}}_0 \in \mathbb{R}^n$ the sequence ${\bar{\mathbf{x}}_k}$ generated by the conjugate direction algorithm converges to the solution $\bar{\mathbf{x}}^*$ of the linear system in at most $n$ steps.

Figures (2)

  • Figure 1: Variation of the norm of the primal residual
  • Figure 2: Variation of the norm of the dual residual

Theorems & Definitions (4)

  • Definition 1: Conjugate Directions
  • Theorem 1
  • Proposition 1
  • Proof