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Newton-PIPG: A Fast Hybrid Algorithm for Quadratic Programs in Optimal Control

Dayou Luo, Yue Yu, Maryam Fazel, Behçet Açıkmeşe

TL;DR

Newton-PIPG presents a fast hybrid solver for structured QPs in optimal control by merging the Proportional-Integral Projected Gradient (PIPG) operator-splitting method with a Newton step. The approach ensures global convergence via PIPG and accelerates convergence locally with a carefully constructed Newton update, under extended-LICQ and strict complementarity. An efficient factorization strategy exploits the typical sparsity of optimal-control QPs to solve the Newton system with reduced complexity. Theoretical guarantees include local nonsingularity of the Newton step and quadratic convergence, while numerical experiments demonstrate substantial speedups over state-of-the-art solvers, especially when feasibility is readily attained. The method shows strong promise for real-time optimal-control applications and can be extended to broader constraint classes and infeasibility detection in future work.

Abstract

We propose Newton-PIPG, an efficient method for solving quadratic programming (QP) problems arising in optimal control, subject to additional set constraints. Newton-PIPG integrates the Proportional-Integral Projected Gradient (PIPG) method with the Newton method, thereby achieving both global convergence and local quadratic convergence. The PIPG method, an operator-splitting algorithm, seeks a fixed point of the PIPG operator. Under mild assumptions, we demonstrate that this operator is locally smooth, enabling the application of the Newton method to solve the corresponding nonlinear fixed-point equation. Furthermore, we prove that the linear system associated with the Newton method is locally nonsingular under strict complementarity conditions. To enhance efficiency, we design a specialized matrix factorization technique that leverages the typical sparsity of optimal control problems in such systems. Numerical experiments demonstrate that Newton-PIPG achieves high accuracy and reduces computation time, particularly when feasibility is easily guaranteed.

Newton-PIPG: A Fast Hybrid Algorithm for Quadratic Programs in Optimal Control

TL;DR

Newton-PIPG presents a fast hybrid solver for structured QPs in optimal control by merging the Proportional-Integral Projected Gradient (PIPG) operator-splitting method with a Newton step. The approach ensures global convergence via PIPG and accelerates convergence locally with a carefully constructed Newton update, under extended-LICQ and strict complementarity. An efficient factorization strategy exploits the typical sparsity of optimal-control QPs to solve the Newton system with reduced complexity. Theoretical guarantees include local nonsingularity of the Newton step and quadratic convergence, while numerical experiments demonstrate substantial speedups over state-of-the-art solvers, especially when feasibility is readily attained. The method shows strong promise for real-time optimal-control applications and can be extended to broader constraint classes and infeasibility detection in future work.

Abstract

We propose Newton-PIPG, an efficient method for solving quadratic programming (QP) problems arising in optimal control, subject to additional set constraints. Newton-PIPG integrates the Proportional-Integral Projected Gradient (PIPG) method with the Newton method, thereby achieving both global convergence and local quadratic convergence. The PIPG method, an operator-splitting algorithm, seeks a fixed point of the PIPG operator. Under mild assumptions, we demonstrate that this operator is locally smooth, enabling the application of the Newton method to solve the corresponding nonlinear fixed-point equation. Furthermore, we prove that the linear system associated with the Newton method is locally nonsingular under strict complementarity conditions. To enhance efficiency, we design a specialized matrix factorization technique that leverages the typical sparsity of optimal control problems in such systems. Numerical experiments demonstrate that Newton-PIPG achieves high accuracy and reduces computation time, particularly when feasibility is easily guaranteed.

Paper Structure

This paper contains 22 sections, 14 theorems, 105 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Road Map
  4. Notation
  5. Background
  6. Proportional-Integral Projected Gradient
  7. Newton Method
  8. Newton Step for PIPG
  9. Extended Linearly Independent Constraint Qualification
  10. Properties of Projections to Convex Sets
  11. Nonsingularity of the Newton Step Matrix
  12. Efficient Computation of Linear System in Newton Step
  13. Global Convergence for Newton-PIPG
  14. Convergence Analysis of Newton-PIPG
  15. Restrictions and Relaxation
  16. ...and 7 more sections

Key Result

Theorem 2.1

Suppose assumptions in Theorem thm: extended-licq hold. Let $(z^k,w^k)$ be computed as equ:iteration, $\alpha, \beta > 0$ and $\alpha\|P\| + \alpha\beta\|H\|^2 < 1$. If $\ \operatorname{Fix}(T)$ is nonempty, the sequence $(z^k, w^k)$ will converge to a point $(z^\star, w^\star) \in \operatorname{Fix

Figures (4)

  • Figure 1: The oscillating masses system
  • Figure 2: Comparison of residuals versus solve time for Newton-PIPG and PIPG algorithms for the oscillating masses example.
  • Figure 3: Comparison of execution times for ECOS, Newton-PIPG, and PIPG with low accuracy (tolerance $10^{-4}$) and high accuracy (tolerance $10^{-8}$) across different initial y locations. Hollow markers represent non-converged points, which are omitted from the legend.
  • Figure 4: Comparison of residuals versus solve time for Newton-PIPG and PIPG algorithms for the PDG example, with $r_\text{init} = (0, 0, 2000)$ meters.

Theorems & Definitions (31)

  • Theorem 2.1
  • Definition 3.1
  • Definition 3.2: Extended-LICQ
  • Theorem 3.3
  • Proof 1
  • Remark 3.4
  • Lemma 3.5
  • Proof 2
  • Theorem 3.6
  • Theorem 3.7
  • ...and 21 more