Exact Quadratic Penalty Function for Symplectic Eigenvalue Problem

TL;DR

This paper presents a novel trace-penalty minimization approach for solving the symplectic eigenvalue problem, transforming it into an unconstrained optimization task and developing an efficient gradient-based algorithm.

Contribution

It introduces a new trace-penalty function method with proven equivalence to the original problem and an efficient gradient algorithm outperforming existing methods.

Findings

The proposed algorithm outperforms Riemannian gradient methods in efficiency.

Numerical experiments show fast convergence for large-scale problems.

The method is effective for dense, sparse, and low-rank matrices.

Abstract

The symplectic eigenvalue problem for symmetric positive-definite (spd) matrices plays a crucial role in various scientific fields, including quantum mechanics and control theory. This paper introduces a trace-penalty minimization method, which transforms the symplectic eigenvalue problem into the unconstrained minimization of the trace-penalty function. We prove the equivalence between the penalty problem and the original constrained optimization problem under mild conditions, in the sense that the second-order stationary points of the trace-penalty function correspond to the solutions of the symplectic eigenvalue problem. Moreover, we develop an algorithm to minimize the trace-penalty function efficiently, which follows the scheme of gradient methods, together with the Barzilai-Borwein (BB) adaptive step-size rule and non-monotone line-search technique. Numerical experiments demonstrate that the proposed algorithm outperforms a wide range of existing methods, such as Riemannian gradient-based methods, in terms of computational efficiency and convergence rate for dense, sparse, and sparse-add-low-rank matrices. These numerical results further demonstrate the great potential of our proposed algorithm, especially in solving large-scale symplectic eigenvalue problems.