A Riemannian symmetric rank-one trust-region method

• 成果类型: Article
• 署名作者: Huang, Wen; Absil, P. -A.; Gallivan, K. A.
• 署名单位: State University System of Florida; Florida State University; Universite Catholique Louvain
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-014-0765-1
• 发表日期: 2015
• 页码: 179-216
• 关键词: quasi-newton matrices; one update; MANIFOLDS; optimization; CONVERGENCE; Tensors

摘要

The well-known symmetric rank-one trust-region method-where the Hessian approximation is generated by the symmetric rank-one update-is generalized to the problem of minimizing a real-valued function over a -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 -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 experiments-Rayleigh quotient minimization on the sphere and a joint diagonalization problem on the Stiefel manifold-illustrate the value of the new methods.