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spopt

A Matlab solver for Riemannian optimization on the symplectic Stiefel manifold

Problems

This solver is to solve the following optimization problem,

  • min f(X), s.t. X' J2n X = J2p

where X is a 2n-by-2p matrix, J2n = [0 In; -In 0], and In is the n-by-n identity matrix.

Applications

  1. the nearest symplectic matrix problem:

min ||X-A||^2_F, s.t. X' J2n X = J2p.

  1. the extrinsic mean problem:

min 1/N \sum_{i=1}^{i=N}||X-A_i||^2_F, s.t. X' J2n X = J2p.

  1. minimization of the Brockett cost function:

min trace(X'AXN-2BX'), s.t. X' J2n X = J2p.

  1. symplectic eigenvalue problem:

min trace(X'AX), s.t. X' J2n X = J2p.

  1. symplectic model order reduction:

min ||M-XX^\dag M||, s.t. X' J2n X = J2p, where X^\dag = J2p' X' J2n

References

Bin Gao, Nguyen Thanh Son, P.-A. Absil, Tatjana Stykel

  1. Riemannian optimization on the symplectic Stiefel manifold, SIAM Journal on Optimization, 31-2 (2021), 1546-1575.
  2. Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric, Geometric Science of Information: 5th International Conference, GSI 2021, Lecture Notes in Computer Science, 12829 (2021), 789–796.
  3. Computing symplectic eigenpairs of symmetric positive-definite matrices via trace minimization and Riemannian optimization, SIAM Journal on Matrix Analysis and Applications, 42-4 (2021), 1732–1757.
  4. Symplectic model order reduction of Hamiltonian systems: a Riemannian optimization approach

Authors

Copyright

Copyright (C) 2020, Bin Gao, Nguyen Thanh Son, P.-A. Absil, Tatjana Stykel

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/

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Riemannian optimization on the symplectic Stiefel manifold

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