Augmented Lagrangian Methods for Optimization on Sp(2p,2n)

Three augmented Lagrangian algorithms for solving optimization problems on the symplectic Stiefel manifold.

Augmented Lagrangian Methods for Optimization on the Symplectic Stiefel Manifold

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1. Problem and solvers

This package contains Matlab codes for solving the following problem

min_{X\in R^{2n\times 2p}} F(X) s.t X'*J_{2n}*X = J_{2p};

where F:R^{2n\times 2p} \to R is a smooth function. The above problema is solved with three augmented lagrangian algoritmos introduced in the preprint [REF]"Augmented Lagrangian Methods for Optimization on the Symplectic Stiefel Manifold", by Oviedo H. and Lara H., (2025).

Solvers: fun.m Grad_cayley.m, Grad_qr.m, Pen_Approach.m, PenGrad_cayley.m, PenGrad_qr.m, RADMM.m

fun.m --- corresponds to an implementation of the objective function and its gradient.

SGM.m --- corresponds to an implementation of the spectral gradient method to

solve every subproblem (to minimize the augmented Lagrangian function).

ALM.m --- corresponds to an implementation of the ALM algorithm presented in

the reference mentioned above [REF].

ALM2.m --- corresponds to an implementation of the ALM2 algorithm presented in

the reference mentioned above [REF].

ALM3.m --- corresponds to an implementation of the ALM3 algorithm presented in

the reference mentioned above [REF].

spopt.m --- corresponds to the algorithms introduced in: Gao B., Son N. T., Absil P. A.

and Stykel T. (2021). Riemannian optimization on the symplectic Stiefel

manifold. SIAM Journal on Optimization, 31(2): 1546–1575. These

methods are available in https://github.com/opt-gaobin/spopt. This code is

not a proprietary implementation, it will be only used for comparison

purposes.

demo.m --- A Matlab script where a trace minimization problem is randomly

generated and solved with all solvers.

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2. Reference

Harry Oviedo and Hugo Larta. "Augmented Lagrangian Methods for Optimization on the Symplectic Stiefel Manifold". Technical report, 2025;

Harry Oviedo <[e-mail]>

Hugo Lara <[e-mail]>

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3. The Authors

We hope that the package is useful for your application. If you have any bug reports or comments, please feel free to email me:

Harry Oviedo, [e-mail]

Enjoy it!

Harry Oviedo and Hugo Lara.

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Copyright (C) 2025, Harry Oviedo and Hugo Lara.

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.