Optimization on the symplectic Stiefel manifold: SR decomposition-based retraction and applications

TL;DR

This paper introduces a novel SR decomposition-based retraction for optimization on the symplectic Stiefel manifold, demonstrating its effectiveness through theoretical proofs and applications in physics and control systems.

Contribution

It proposes a new retraction method based on SR matrix decomposition and proves its domain contains the open unit ball, enabling global convergence analysis.

Findings

The new retraction is effective for symplectic optimization problems.

Numerical experiments show improved performance over existing methods.

Applications include symplectic eigenvalue computation and Hamiltonian system reduction.

Abstract

Numerous problems in optics, quantum physics, stability analysis, and control of dynamical systems can be brought to an optimization problem with matrix variable subjected to the symplecticity constraint. As this constraint nicely forms a so-called symplectic Stiefel manifold, Riemannian optimization is preferred, because one can borrow ideas from unconstrained optimization methods after preparing necessary geometric tools. Retraction is arguably the most important one which decides the way iterates are updated given a search direction. Two retractions have been constructed so far: one relies on the Cayley transform and the other is designed using quasi-geodesic curves. In this paper, we propose a new retraction which is based on an SR matrix decomposition. We prove that its domain contains the open unit ball which is essential in proving the global convergence of the associated gradient-based optimization algorithm. Moreover, we consider three applications--symplectic target matrix problem, symplectic eigenvalue computation, and symplectic model reduction of Hamiltonian systems--with various examples. The extensive numerical comparisons reveal the strengths of the proposed optimization algorithm.