A Riemannian symmetric rank-one trust-region method
The well-known symmetric rank-one trust-region methodwhere the Hessian approximation is generated by the symmetric rank-one updateis generalized to the problem of minimizing a real-valued function over a \(d\)-dimensional Riemannian manifold. The generalization relies on basic differential-geometric concepts, such as tangent spaces, Riemannian metrics, and the Riemannian gradient, as well as on the more recent notions of (first-order) retraction and vector transport. The new method, called RTR-SR1, is shown to converge globally and \(d+1\)-step q-superlinearly to stationary points of the objective function. A limited-memory version, referred to as LRTR-SR1, is also introduced. In this context, novel efficient strategies are presented to construct a vector transport on a submanifold of a Euclidean space. Numerical experimentsRayleigh quotient minimization on the sphere and a joint diagonalization problem on the Stiefel manifoldillustrate the value of the new methods.
joint diagonalization riemannian optimization optimization on manifolds symmetric rank-one update rayleigh quotient stiefel manifold 65k05 90c48 90c53