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Optimization methods for solving matrix equations

Juan Zhang, Xiao Luo

TL;DR

This work reframes the Sylvester equation and the continuous algebraic Riccati equation (CARE) as optimization problems and compares several optimization-based solvers. It shows that a constrained convex approach using the $\ell_{2,1}$-norm (CCOM) yields global convergence for Sylvester solves but is not scalable, while classical quasi-Newton methods (DFP, BFGS) with Wolfe-Powell line search provide robust, efficient Sylvester solutions. For CARE, the study develops a unified ADMM framework and then couples it with a Newton method (Newton-ADMM) to achieve superior efficiency, demonstrated by numerical experiments. Overall, the paper highlights a spectrum of optimization-based strategies for matrix equations, with Newton-ADMM offering a practical, scalable path to fast CARE solutions, albeit with sensitivity to algorithmic parameters that warrant further tuning and study.

Abstract

In this paper, we focus on using optimization methods to solve matrix equations by transforming the problem of solving the Sylvester matrix equation or continuous algebraic Riccati equation into an optimization problem. Initially, we use a constrained convex optimization method (CCOM) to solve the Sylvester matrix equation with $\ell_{2,1}$-norm, where we provide a convergence analysis and numerical examples of CCOM; however, the results show that the algorithm is not efficient. To address this issue, we employ classical quasi-Newton methods such as DFP and BFGS algorithms to solve the Sylvester matrix equation and present the convergence and numerical results of the algorithm. Additionally, we compare these algorithms with the CG algorithm and AR algorithm, and our results demonstrate that the presented algorithms are effective. Furthermore, we propose a unified framework of the alternating direction multiplier method (ADMM) for directly solving the continuous algebraic Riccati equation (CARE), and we provide the convergence and numerical results of ADMM. Our experimental results indicate that ADMM is an effective optimization algorithm for solving CARE. Finally, to improve the effectiveness of the optimization method for solving Riccati equation, we propose the Newton-ADMM algorithm framework, where the outer iteration of this method is the classical Newton method, and the inner iteration involves using ADMM to solve Lyapunov matrix equations inexactly. We also provide the convergence and numerical results of this algorithm, which our results demonstrate are more efficient than ADMM for solving CARE.

Optimization methods for solving matrix equations

TL;DR

This work reframes the Sylvester equation and the continuous algebraic Riccati equation (CARE) as optimization problems and compares several optimization-based solvers. It shows that a constrained convex approach using the -norm (CCOM) yields global convergence for Sylvester solves but is not scalable, while classical quasi-Newton methods (DFP, BFGS) with Wolfe-Powell line search provide robust, efficient Sylvester solutions. For CARE, the study develops a unified ADMM framework and then couples it with a Newton method (Newton-ADMM) to achieve superior efficiency, demonstrated by numerical experiments. Overall, the paper highlights a spectrum of optimization-based strategies for matrix equations, with Newton-ADMM offering a practical, scalable path to fast CARE solutions, albeit with sensitivity to algorithmic parameters that warrant further tuning and study.

Abstract

In this paper, we focus on using optimization methods to solve matrix equations by transforming the problem of solving the Sylvester matrix equation or continuous algebraic Riccati equation into an optimization problem. Initially, we use a constrained convex optimization method (CCOM) to solve the Sylvester matrix equation with -norm, where we provide a convergence analysis and numerical examples of CCOM; however, the results show that the algorithm is not efficient. To address this issue, we employ classical quasi-Newton methods such as DFP and BFGS algorithms to solve the Sylvester matrix equation and present the convergence and numerical results of the algorithm. Additionally, we compare these algorithms with the CG algorithm and AR algorithm, and our results demonstrate that the presented algorithms are effective. Furthermore, we propose a unified framework of the alternating direction multiplier method (ADMM) for directly solving the continuous algebraic Riccati equation (CARE), and we provide the convergence and numerical results of ADMM. Our experimental results indicate that ADMM is an effective optimization algorithm for solving CARE. Finally, to improve the effectiveness of the optimization method for solving Riccati equation, we propose the Newton-ADMM algorithm framework, where the outer iteration of this method is the classical Newton method, and the inner iteration involves using ADMM to solve Lyapunov matrix equations inexactly. We also provide the convergence and numerical results of this algorithm, which our results demonstrate are more efficient than ADMM for solving CARE.
Paper Structure (37 sections, 22 theorems, 172 equations, 10 tables, 5 algorithms)

This paper contains 37 sections, 22 theorems, 172 equations, 10 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. CCOM Algorithm
  3. Iterative Method
  4. Convergent Analysis
  5. Experimental Results
  6. Example 1
  7. Example 2
  8. Analysis of numerical results
  9. Quasi-Newton Method
  10. Preliminaries
  11. Iterative Methods
  12. DFP Algorithm
  13. BFGS Algorithm
  14. Convergent Analysis
  15. Convergence of Algorithm \ref{['alg4']}
  16. ...and 22 more sections

Key Result

Lemma 1

For any nonzero vectors $u,u_t \in \mathbb{R}^c$, the following inequality holds:

Theorems & Definitions (26)

  • Lemma 1
  • Theorem 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Theorem 2
  • Lemma 7
  • ...and 16 more