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Blocked Cholesky factorization updates of the Riccati recursion using hyperbolic Householder transformations

Pieter Pas, Panagiotis Patrinos

TL;DR

The paper addresses the cost of solving sequential Newton systems in QP solvers by updating Cholesky factorizations under low-rank nudges, rather than recomputing them. It introduces hyperbolic Householder transformations, encapsulating updates via $\breve Q$ and enabling efficient, blocked factorization updates for matrices of the form $LL^{\top} + A\Sigma A^{\top}$. By applying this framework to the Riccati recursion inherent in augmented Lagrangian-based OCPs, the authors derive a novel, efficient update procedure for the Cholesky factors across Riccati stages, with a rigorous induction showing how the cost-to-go matrices remain low-rank perturbed. An optimized open-source C++ implementation (HyHound) demonstrates substantial runtime reductions and superior performance compared to full factorization and Schur-complement approaches, highlighting practical impact for real-time optimal control solvers.

Abstract

Newton systems in quadratic programming (QP) methods are often solved using direct Cholesky or LDL factorizations. When the linear systems in successive iterations differ by a low-rank modification (as is common in active set and augmented Lagrangian methods), updating the existing factorization can offer significant performance improvements over recomputing a full Cholesky factorization. We review the hyperbolic Householder transformation, and demonstrate its usefulness in describing low-rank Cholesky factorization updates. By applying this hyperbolic Householder-based framework to the well-known Riccati recursion for solving saddle-point problems with optimal control structure, we develop a novel algorithm for updating the factorizations used in optimization solvers for optimal control. Specifically, the proposed method can be used to efficiently solve the semismooth Newton systems that are at the core of the augmented Lagrangian-based QPALM-OCP solver. An optimized open-source implementation of the proposed factorization update routines is provided as well.

Blocked Cholesky factorization updates of the Riccati recursion using hyperbolic Householder transformations

TL;DR

The paper addresses the cost of solving sequential Newton systems in QP solvers by updating Cholesky factorizations under low-rank nudges, rather than recomputing them. It introduces hyperbolic Householder transformations, encapsulating updates via and enabling efficient, blocked factorization updates for matrices of the form . By applying this framework to the Riccati recursion inherent in augmented Lagrangian-based OCPs, the authors derive a novel, efficient update procedure for the Cholesky factors across Riccati stages, with a rigorous induction showing how the cost-to-go matrices remain low-rank perturbed. An optimized open-source C++ implementation (HyHound) demonstrates substantial runtime reductions and superior performance compared to full factorization and Schur-complement approaches, highlighting practical impact for real-time optimal control solvers.

Abstract

Newton systems in quadratic programming (QP) methods are often solved using direct Cholesky or LDL factorizations. When the linear systems in successive iterations differ by a low-rank modification (as is common in active set and augmented Lagrangian methods), updating the existing factorization can offer significant performance improvements over recomputing a full Cholesky factorization. We review the hyperbolic Householder transformation, and demonstrate its usefulness in describing low-rank Cholesky factorization updates. By applying this hyperbolic Householder-based framework to the well-known Riccati recursion for solving saddle-point problems with optimal control structure, we develop a novel algorithm for updating the factorizations used in optimization solvers for optimal control. Specifically, the proposed method can be used to efficiently solve the semismooth Newton systems that are at the core of the augmented Lagrangian-based QPALM-OCP solver. An optimized open-source implementation of the proposed factorization update routines is provided as well.

Paper Structure

This paper contains 9 sections, 4 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hyperbolic Householder transformations
  3. Construction and properties
  4. Efficient representation and application of hyperbolic Householder transformations
  5. Algorithms for factorization updates
  6. Two core algorithms, two perspectives
  7. Blocked algorithm
  8. Application to the Riccati recursion
  9. Performance of optimized updating routines

Key Result

Lemma 1

Consider a matrix $\mathscr S \triangleq \left(\! \!\right) \in \mathbb{D}(\mathrm{I\space R}^{n})$ and a vector $x \in \mathrm{I\space R}^{n}$ with $x^{\!\top\!} \mathscr S x > 0$. Then the matrix $\breve Q \in \mathrm{I\space R}^{n\times n}$ defined below is $\mathscr S$-orthogonal, and satisfies

Figures (2)

  • Figure 1: Absolute run time of different factorization update methods, for $L\in \mathrm{I\space R}^{64\times64}, A\in \mathrm{I\space R}^{64\times m}$. Full factorization refers to adding $A\Sigma A^{\!\top\!}$ to the previous matrix $H$ (dsyrk) and performing a fresh Cholesky factorization (dpotrf). HyH update refers to the blocked hyperbolic Householder-based algorithm from §\ref{['sec:blocked-alg']}, for various block sizes $r$.
  • Figure 2: Run time comparison of the individual factorization, solution, and factorization update steps when solving \ref{['eq:ocp-qpalm-inner']}, for the Schur complement method versus the Riccati recursion from §\ref{['sec:riccati']}. OCP dimensions: $N\!=\!24,n_x\!=\!24, n_u\!=\!8$.

Theorems & Definitions (14)

  • Definition 1: $\mathscr S$-orthogonality
  • Lemma 1: Hyperbolic Householder reflectors
  • proof
  • Remark 1: Hyperbolic Householder matrix construction
  • Remark 2: Alternative definitions
  • Proposition 1: Hyperbolic $L \breve Q$ factorization
  • proof
  • Corollary 1: Factorization updates
  • proof
  • Lemma 2: Product of hyperbolic Householder matrices
  • ...and 4 more