On computing the distance to stability for matrices using linear dissipative Hamiltonian systems

TL;DR

This paper introduces new algorithms based on linear dissipative Hamiltonian systems to efficiently compute the nearest stable matrix to an unstable one, offering improved performance over existing methods.

Contribution

The paper reformulates the stability problem using a convex optimization approach with novel algorithms, including a fast gradient method, for better computational efficiency.

Findings

The fast gradient method outperforms other algorithms in speed and accuracy.

The reformulation simplifies the stability problem into a convex optimization task.

The proposed algorithms require $\\mathcal{O}(n^3)$ operations per iteration.

Abstract

In this paper, we consider the problem of computing the nearest stable matrix to an unstable one. We propose new algorithms to solve this problem based on a reformulation using linear dissipative Hamiltonian systems: we show that a matrix $A$ is stable if and only if it can be written as $A = (J-R)Q$, where $J=-J^T$, $R \succeq 0$ and $Q \succ 0$ (that is, $R$ is positive semidefinite and $Q$ is positive definite). This reformulation results in an equivalent optimization problem with a simple convex feasible set. We propose three strategies to solve the problem in variables $(J,R,Q)$: (i) a block coordinate descent method, (ii) a projected gradient descent method, and (iii) a fast gradient method inspired from smooth convex optimization. These methods require $\mathcal{O}(n^3)$ operations per iteration, where $n$ is the size of $A$. We show the effectiveness of the fast gradient method compared to the other approaches and to several state-of-the-art algorithms.