Optimization on the real symplectic group. (English) Zbl 1434.49026
Summary: We regard the real symplectic group \(Sp(2n,{{\mathbb{R}}})\) as a constraint submanifold of the \(2n\times 2n\) real matrices \({\mathcal{M}}_{2n}({{\mathbb{R}}})\) endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of the general linear group \(Gl(2n,{{\mathbb{R}}})\) endowed with the (left) invariant metric. For a cost function that defines an optimization problem on the real symplectic group we give a necessary and sufficient condition for critical points and we apply this condition to the particular case of a least square cost function. In order to characterize the critical points we give a formula for the Hessian of a cost function defined on the real symplectic group, with respect to both considered metrics. For a generalized Brockett cost function we present a necessary condition and a sufficient condition for local minimum. We construct a retraction map that allows us to detail the steepest descent and embedded Newton algorithms for solving an optimization problem on the real symplectic group.
MSC:
| 49M15 | Newton-type methods |
| 15A99 | Basic linear algebra |
| 53C30 | Differential geometry of homogeneous manifolds |
Keywords:
optimization; constraint manifold; real symplectic group; retraction map; steepest descent algorithm; Newton algorithmReferences:
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