Strong Duality Thresholds and Minimum Curvature for the Nearest Rectangular Symplectic Matrix Problem
Abstract
We study the nearest rectangular symplectic matrix problem in the squared Frobenius norm for arbitrary targets T∈ℝ2n×2k, with n≥ k. We prove that its semidefinite relaxation is exact for every target if and only if n≥2k−1. Under this condition, the primal problem, the relaxation, and both Lagrangian dual problems attain the same optimal value. In this range, we derive the exact minimum Rayleigh quotient of the Riemannian Hessian at every feasible point under the Euclidean induced metric, and prove that all second-order stationary points are global minimizers. We also obtain global objective error bounds. We extend the strong-duality and minimum-curvature results to Hermitian positive definite row weights, proving that strong duality holds for every target if and only if the lowest eigenvalue of the weight has complex multiplicity at least 2k−1. The dual formulation leads to a method whose main iterations use matrices of order 2k, with O(k2) iterative storage apart from the full input and output. Across six tested sizes, mean-time speedups are $2.55−−12.60$ over a Riemannian first-order method and $4.61−−43.93$ over Jensen--Zimmermann R--TR2; batching gives a further factor of $3.45−−23.53$ over sequential calls. The theoretical results complete the critical dimensions not covered by Gram convexification and quadratic matrix programming.
Related
- Minimum b2 of symplectic fillings of lens spaces
- Sharp criteria for the existence of solutions to curvature equations with two singular sources on rectangular torus
- Symplectic singularities in Relative Langlands duality for products of general linear groups
- Minimum-degree thresholds beyond bounded colourability
- Fedosov manifolds with parallel symplectic Weyl curvature
BibTeX
@misc{xie2026strong,
title = {{Strong Duality Thresholds and Minimum Curvature for the Nearest Rectangular Symplectic Matrix Problem}},
author = {Yulin Xie and Aiqing Zhu and Yifa Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31298},
eprint = {2610.31298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31298},
}
RIS
TY - UNPB
TI - Strong Duality Thresholds and Minimum Curvature for the Nearest Rectangular Symplectic Matrix Problem
AU - Yulin Xie
AU - Aiqing Zhu
AU - Yifa Tang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31298
UR - https://arxiv.org/abs/2610.31298
AB - We study the nearest rectangular symplectic matrix problem in the squared Frobenius norm for arbitrary targets T∈ℝ^2n×2k, with n≥ k. We prove that its semidefinite relaxation is exact for every target if and only if n≥2k−1. Under this condition, the primal problem, the relaxation, and both Lagrangian dual problems attain the same optimal value. In this range, we derive the exact minimum Rayleigh quotient of the Riemannian Hessian at every feasible point under the Euclidean induced metric, and prove that all second-order stationary points are global minimizers. We also obtain global objective error bounds. We extend the strong-duality and minimum-curvature results to Hermitian positive definite row weights, proving that strong duality holds for every target if and only if the lowest eigenvalue of the weight has complex multiplicity at least 2k−1. The dual formulation leads to a method whose main iterations use matrices of order 2k, with O(k^2) iterative storage apart from the full input and output. Across six tested sizes, mean-time speedups are $2.55−−12.60$ over a Riemannian first-order method and $4.61−−43.93$ over Jensen--Zimmermann R--TR2; batching gives a further factor of $3.45−−23.53$ over sequential calls. The theoretical results complete the critical dimensions not covered by Gram convexification and quadratic matrix programming.
ER -