The symplectic Stiefel manifold, denoted by ${Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. Optimization problems on ${Sp}(2p,2n)$ find applications in various areas, such as optics, quantum physics, numerical linear algebra, and model order reduction of dynamical systems. The purpose of this paper is to propose and analyze gradient-descent methods on ${Sp}(2p,2n)$, where the notion of gradient stems from a Riemannian metric. We consider a novel Riemannian metric on ${Sp}(2p,2n)$ akin to the canonical metric of the (standard) Stiefel manifold. In order to perform a feasible step along the antigradient, we develop two types of search strategies: one is based on quasi-geodesic curves and the other one on the symplectic Cayley transform. The resulting optimization algorithms are proved to converge globally to critical points of the objective function. Numerical experiments illustrate the efficiency of the proposed methods.

  • 1.  P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2008. Google Scholar

  • 2.  R. L. Adler J.-P. Dedieu J. Y. Margulies M. Martens and  M. Shub , Newton's method on Riemannian manifolds and a geometric model for the human spine , IMA J. Numer. Anal. , 22 ( 2002 ), pp. 359 -- 390 , https://doi.org/10.1093/imanum/22.3.359. CrossrefISIGoogle Scholar

  • 3.  B. Afkham and  J. Hesthaven , Structure preserving model of parametric Hamiltonian systems , SIAM J. Sci. Comput. , 39 ( 2017 ), pp. A2616 -- A2644 , https://doi.org/10.1137/17M1111991. LinkISIGoogle Scholar

  • 4.  D. O. A. Ajayi and  A. Banyaga , An explicit retraction of symplectic flag manifolds onto complex flag manifolds , J. Geometry , 104 ( 2013 ), pp. 1 -- 9 , https://doi.org/10.1007/s00022-013-0148-4. CrossrefGoogle Scholar

  • 5.  J. Barzilai and  J. M. Borwein , Two-point step size gradient methods , IMA J. Numer. Anal. , 8 ( 1988 ), pp. 141 -- 148 , https://doi.org/10.1093/imanum/8.1.141. CrossrefISIGoogle Scholar

  • 6.  D. P. Bertsekas, Nonlinear Programming, Athena Scientific, Nashua, NH, 1995, http://www.athenasc.com/nonlinbook.html. Google Scholar

  • 7.  R. Bhatia and  T. Jain , On symplectic eigenvalues of positive definite matrices , J. Math. Phys. , 56 ( 2015 ), p. 112201 , https://doi.org/10.1063/1.4935852. CrossrefISIGoogle Scholar

  • 8.  A. Bhattacharya and R. Bhattacharya, Nonparametric Inference on Manifolds: With Applications to Shape Spaces, Cambridge University Press, Cambridge, UK, 2012, https://doi.org/10.1017/CBO9781139094764. Google Scholar

  • 9.  R. Bhattacharya and  V. Patrangenaru , Large sample theory of intrinsic and extrinsic sample means on manifolds , Ann. Statist. , 31 ( 2003 ), pp. 1 -- 29 , https://doi.org/10.1214/aos/1046294456. CrossrefISIGoogle Scholar

  • 10.  P. Birtea I. Caşu and  D. Comănescu , Optimization on the real symplectic group , Monatsh. Math. , 191 ( 2020 ), pp. 465 -- 485 , https://doi.org/10.1007/s00605-020-01369-9. CrossrefISIGoogle Scholar

  • 11.  N. Boumal P.-A. Absil and  C. Cartis , Global rates of convergence for nonconvex optimization on manifolds , IMA J. Numer. Anal. , 39 ( 2018 ), pp. 1 -- 33 , https://doi.org/10.1093/imanum/drx080. CrossrefISIGoogle Scholar

  • 12.  N. Boumal B. Mishra P.-A. Absil and  R. Sepulchre , Manopt, a MATLAB toolbox for optimization on manifolds , J. Mach. Learn. Res. , 15 ( 2014 ), pp. 1455 -- 1459 , https://www.manopt.org. ISIGoogle Scholar

  • 13.  R. W. Brockett , Least squares matching problems , Linear Algebra Appl. , 122-124 ( 1989 ), pp. 761 -- 777 , https://doi.org/10.1016/0024-3795(89)90675-7. CrossrefISIGoogle Scholar

  • 14.  P. Buchfink A. Bhatt and  B. Haasdonk , Symplectic model order reduction with non-orthonormal bases , Math. Comput. Appl. , 24 ( 2019 ), https://doi.org/ 10 .3390/mca24020043. Google Scholar

  • 15.  Y.-H. Dai and  R. Fletcher , Projected Barzilai--Borwein methods for large-scale box-constrained quadratic programming , Numer. Math. , 100 ( 2005 ), pp. 21 -- 47 , https://doi.org/10.1007/s00211-004-0569-y. CrossrefISIGoogle Scholar

  • 16.  M. A. de Gosson, Symplectic Geometry and Quantum Mechanics, Adv. Partial Differ. Equ. (Basel) 166, Springer, Basel, 2006, https://doi.org/10.1007/3-7643-7575-2. Google Scholar

  • 17.  M. . de Gosson and F. Luef, Metaplectic group, symplectic Cayley transform, and fractional Fourier transforms , J. Math. Anal. Appl. , 416 ( 2014 ), pp. 947 -- 968 , https://doi.org/10.1016/j.jmaa.2014.03.013. CrossrefISIGoogle Scholar

  • 18.  A. Draft F. Neri G. Rangarajan D. R. Douglas L. M. Healy and  R. D. Ryne , Lie algebraic treatment of linear and nonlinear beam dynamics , Ann. Rev. Nuclear Particle Sci. , 38 ( 1988 ), pp. 455 -- 496 , https://doi.org/10.1146/annurev.ns.38.120188.002323. CrossrefISIGoogle Scholar

  • 19.  A. Edelman T. A. Arias and  S. T. Smith , The geometry of algorithms with orthogonality constraints , SIAM J. Matrix Anal. Appl. , 20 ( 1998 ), pp. 303 -- 353 , https://doi.org/10.1137/S0895479895290954. LinkISIGoogle Scholar

  • 20.  S. Fiori , Solving minimal-distance problems over the manifold of real-symplectic matrices , SIAM J. Matrix Anal. Appl. , 32 ( 2011 ), pp. 938 -- 968 , https://doi.org/10.1137/100817115. LinkISIGoogle Scholar

  • 21.  S. Fiori Riemannian steepest descent approach over the inhomogeneous symplectic group: Application to the averaging of linear optical systems , Appl. Math. Comput. , 283 ( 2016 ), pp. 251 -- 264 , https://doi.org/10.1016/j.amc.2016.02.018. CrossrefISIGoogle Scholar

  • 22.  G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013. Google Scholar

  • 23.  E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer Science & Business Media, New York, 2006, https://doi.org/10.1007/3-540-30666-8. Google Scholar

  • 24.  W. Harris , The average eye , Opthalmic Physiol. Optics , 24 ( 2004 ), pp. 580 -- 585 , https://doi.org/10.1111/j.1475-1313.2004.00239.x. CrossrefISIGoogle Scholar

  • 25.  T. Hiroshima , Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs , Phys. Rev. A , 73 ( 2006 ), 012330 , https://doi.org/10.1103/PhysRevA.73.012330. CrossrefISIGoogle Scholar

  • 26.  W.-y. Hsiang and  J. Su , On the classification of transitive effective actions on Stiefel manifolds , Trans. Amer. Math. Soc. , 130 ( 1968 ), pp. 322 -- 336 , https://doi.org/10.1090/S0002-9947-1968-0221529-1. CrossrefISIGoogle Scholar

  • 27.  J. Hu X. Liu and  Z.-W. Wen . Yuan, A brief introduction to manifold optimization , J. Oper. Res. Soc. China , 8 ( 2020 ), pp. 199 -- 243 , https://doi.org/10.1007/s40305-020-00295-9. CrossrefISIGoogle Scholar

  • 28.  W. Huang P.-A. Absil and  K. Gallivan , A Riemannian BFGS method without differentiated retraction for nonconvex optimization problems , SIAM J. Optim. , 28 ( 2018 ), pp. 470 -- 495 , https://doi.org/10.1137/17M1127582. LinkISIGoogle Scholar

  • 29.  B. Iannazzo and  M. Porcelli , The Riemannian Barzilai--Borwein method with nonmonotone line search and the matrix geometric mean computation , IMA J. Numer. Anal. , 38 ( 2018 ), pp. 495 -- 517 , https://doi.org/10.1093/imanum/drx015. CrossrefISIGoogle Scholar

  • 30.  L. M. Machado and F. S. Leite, Optimization on Quadratic Matrix Lie Groups, tech. report, Centro de Mathemática da Universidade de Coimbra, 2002, http://hdl.handle.net/10316/11446. Google Scholar

  • 31.  K. R. Meyer and  G. R. Hall , Linear Hamiltonian Systems, Springer , New York , 1992 , pp. 33 -- 71 , https://doi.org/10.1007/978-1-4757-4073-8_2. Google Scholar

  • 32.  J. Nocedal and S. Wright, Numerical Optimization, Springer Science & Business Media, New York, 2006, https://doi.org/10.1007/978-0-387-40065-5. Google Scholar

  • 33.  L. Peng and  K. Mohseni , Symplectic model reduction of Hamiltonian systems , SIAM J. Sci. Comput. , 38 ( 2016 ), pp. A1 -- A27 , https://doi.org/10.1137/140978922. LinkISIGoogle Scholar

  • 34.  W. Ring and  B. Wirth , Optimization methods on Riemannian manifolds and their application to shape space , SIAM J. Optim. , 22 ( 2012 ), pp. 596 -- 627 , https://doi.org/10.1137/11082885X. LinkISIGoogle Scholar

  • 35.  F. Sigrist and  U. Suter , Cross-sections of symplectic Stiefel manifolds , Trans. Amer. Math. Soc. , 184 ( 1973 ), pp. 247 -- 259 , https://doi.org/10.1090/S0002-9947-1973-0326728-8. CrossrefISIGoogle Scholar

  • 36.  N. T. Son, P.-A. Absil, B. Gao, and T. Stykel, Symplectic Eigenvalue Problem Via Trace Minimization and Riemannian Optimization, arXiv preprint: 2101.02618 [mat. OC], 2021, https://arxiv.org/abs/2101.02618. Google Scholar

  • 37.  J. Wang H. Sun and  S. Fiori , A Riemannian-steepest-descent approach for optimization on the real symplectic group, Math. methods Appl . Sci. , 41 ( 2018 ), pp. 4273 -- 4286 , https://doi.org/10.1002/mma.4890. Google Scholar

  • 38.  Z. Wen and  W. Yin , A feasible method for optimization with orthogonality constraints , Math. Program. , 142 ( 2013 ), pp. 397 -- 434 , https://doi.org/10.1007/s10107-012-0584-1. CrossrefISIGoogle Scholar

  • 39.  J. Williamson , On the algebraic problem concerning the normal forms of linear dynamical systems , Amer. J. Math. , 58 ( 1936 ), pp. 141 -- 163 , https://doi.org/10.2307/2371062. CrossrefGoogle Scholar

  • 40.  R. Wu R. Chakrabarti and  H. Rabitz , Optimal control theory for continuous-variable quantum gates , Phys. Rev. A , 77 ( 2008 ), 052303 , https://doi.org/10.1103/PhysRevA.77.052303. CrossrefISIGoogle Scholar

  • 41.  R.-B. Wu R. Chakrabarti and  H. Rabitz , Critical landscape topology for optimization on the symplectic group , J. Optim. Theory Appl. , 145 ( 2010 ), pp. 387 -- 406 , https://doi.org/10.1007/s10957-009-9641-1. CrossrefISIGoogle Scholar

  • 42.  H. Zhang and  W. W. Hager , A nonmonotone line search technique and its application to unconstrained optimization , SIAM J. Optim. , 14 ( 2004 ), pp. 1043 -- 1056 , https://doi.org/10.1137/S1052623403428208. LinkISIGoogle Scholar