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Optimal Preconditioning for Online Quadratic Cone Programming

Abhinav G. Kamath, Purnanand Elango, Behçet Açıkmeşe

TL;DR

This paper tackles online quadratic cone programs with strongly convex objectives by introducing hypersphere preconditioning, a three-step procedure that preserves problem structure while significantly improving conditioning. The method consists of (1) hypersphere objective preconditioning to achieve unit Hessian conditioning, (2) block row-normalization of the constraint matrix, and (3) an analytically derived optimal objective scaling that minimizes the KKT conditioning, with the only iterative piece being shifted power iteration. The authors derive closed-form expressions for the optimal scaling and provide a lower bound on the resulting KKT conditioning, demonstrating improved performance on convex optimal-control problems and nonconvex sequential conic optimization in online settings; they compare against Ruiz equilibration and a QR preconditioner and validate the approach on trajectory optimization tasks, including SeCO-based rocket guidance. The results indicate that hypersphere preconditioning, particularly when combined with optimal scaling, substantially reduces convergence effort for first-order conic solvers and enables efficient online re-solving in real time, making it well-suited for MPC and sequential optimization tasks. The work also provides practical insights for estimating key spectral quantities via shifted power iteration, maintaining a mostly analytical framework that avoids costly factorizations.

Abstract

First-order conic optimization solvers are sensitive to problem conditioning and typically perform poorly in the face of ill-conditioned problem data. To mitigate this, we propose an approach to preconditioning--the hypersphere preconditioner--for a class of quadratic cone programs (QCPs), i.e., conic optimization problems with a quadratic objective function, wherein the objective function is strongly convex and possesses a certain structure. This approach lends itself to factorization-free, customizable, first-order conic optimization for online applications wherein the solver is called repeatedly to solve problems of the same size/structure, but with changing problem data. We demonstrate the efficacy of our approach on numerical convex and nonconvex trajectory optimization examples, using a first-order conic optimizer under the hood.

Optimal Preconditioning for Online Quadratic Cone Programming

TL;DR

This paper tackles online quadratic cone programs with strongly convex objectives by introducing hypersphere preconditioning, a three-step procedure that preserves problem structure while significantly improving conditioning. The method consists of (1) hypersphere objective preconditioning to achieve unit Hessian conditioning, (2) block row-normalization of the constraint matrix, and (3) an analytically derived optimal objective scaling that minimizes the KKT conditioning, with the only iterative piece being shifted power iteration. The authors derive closed-form expressions for the optimal scaling and provide a lower bound on the resulting KKT conditioning, demonstrating improved performance on convex optimal-control problems and nonconvex sequential conic optimization in online settings; they compare against Ruiz equilibration and a QR preconditioner and validate the approach on trajectory optimization tasks, including SeCO-based rocket guidance. The results indicate that hypersphere preconditioning, particularly when combined with optimal scaling, substantially reduces convergence effort for first-order conic solvers and enables efficient online re-solving in real time, making it well-suited for MPC and sequential optimization tasks. The work also provides practical insights for estimating key spectral quantities via shifted power iteration, maintaining a mostly analytical framework that avoids costly factorizations.

Abstract

First-order conic optimization solvers are sensitive to problem conditioning and typically perform poorly in the face of ill-conditioned problem data. To mitigate this, we propose an approach to preconditioning--the hypersphere preconditioner--for a class of quadratic cone programs (QCPs), i.e., conic optimization problems with a quadratic objective function, wherein the objective function is strongly convex and possesses a certain structure. This approach lends itself to factorization-free, customizable, first-order conic optimization for online applications wherein the solver is called repeatedly to solve problems of the same size/structure, but with changing problem data. We demonstrate the efficacy of our approach on numerical convex and nonconvex trajectory optimization examples, using a first-order conic optimizer under the hood.
Paper Structure (11 sections, 6 theorems, 27 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 11 sections, 6 theorems, 27 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization
  4. Preconditioning
  5. Objective function: hypersphere preconditioning
  6. Constraints: block row-normalization
  7. Optimal objective function scaling factor
  8. Numerical results
  9. Conclusion
  10. Optimal step-size ratio in PIPG
  11. Shifted power iteration

Key Result

Lemma 1

For a given $\lambda > 0$ and $H \in \mathbb{R}^{m \times n}$, where $n > m$ and $\rank H = m$, the spectrum of $K$ is: where $\sigma_{1},\,\ldots,\,\sigma_{m}$ are the squares of the singular values of $H$. The eigenvalue $\lambda$ is repeated $n-m$ times, and there are a total of $n + m$ eigenvalues.

Figures (4)

  • Figure 1: The minimizer of the maximum of a strictly increasing function, $f(x)$, and a strictly decreasing function, $g(x)$, satisfies $f(x) = g(x)$.
  • Figure 2: The condition number of the preconditioned KKT matrix as a function of the terminal state cost matrix scaling factor, $\gamma$. "None" refers to the case where no preconditioning is applied.
  • Figure 3: The combined solve time (preconditioning and convex solve) as a function of $\gamma$. The combined solve time is only plotted if pipg was able to converge (within $10^{5}$ iterations).
  • Figure 4: The effect of the optimal objective function scaling factor, $\lambda^{\star}$—on (i) the condition number, $\kappa(\lambda)$, of the KKT matrix, and (ii) the number of pipg iterations—at each iteration of SeCO for the multi-phase rocket landing guidance problem kamath2023real.

Theorems & Definitions (13)

  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Theorem 2
  • proof
  • ...and 3 more