Optimization on the real symplectic group

Abstract

We regard the real symplectic group Sp(2n, R) as a constraint submanifold of the 2n × 2n real matrices M2n(R) endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of the general linear group Gl(2n, 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.

Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric

“…In the case of the symplectic group Sp(2n), the Riemannian gradient ( 7) is equivalent to the formulation in [8], where the minimization problem was treated as a constrained optimization problem in the Euclidean space. We notice that Ω X in ( 7) is actually the Lagrangian multiplier of the symplectic constraints; see [8].…”

“…In recent decades, most of studies on the symplectic topic focused on the symplectic group (p = n) including geodesics of the symplectic group [10], optimality conditions for optimization problems on the symplectic group [15,20,8], and optimization algorithms on the symplectic group [11,18]. However, there was less attention to the geometry of the symplectic Stiefel manifold Sp(2p, 2n).…”

Unit Triangular Factorization of the Matrix Symplectic Group

“…There have been many works on optimization on the real symplectic group [5,11,27], in which one performs optimization by considering the gradients along the manifold. [19] has pointed out that the unit triangular factorization provides an approach to the symplectic optimization from a new perspective, i.e., optimizing in a higher dimensional unconstrained parameter space.…”

Lyapunov matrix equation $A^HX+XA+C=O_r$ with $A$ a Jordan matrix

“…The Lyapunov matrix equation appeared, see [11] and [7], in the study of the dynamics of a linear differential equation, ẋ = Ax, where A ∈ M r (C). Dynamical aspects, like stability, are studied using a square function V (x) = 1 2 x H Xx, with X ∈ M r (C) a Hermitian matrix. The function V is a conserved quantity of the linear system if and only if X is a Hermitian solution of the homogeneous Lyapunov matrix equation (1.2).…”

“…Lyapunov matrix equations appear also in the study of optimization of a cost function defined on a matrix manifold, see [1].…”