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SIAM J. NUMER. ANAL. c 2009 Society for Industrial and Applied Mathematics Vol. 47, No. 2, pp. 9971018
 

Summary: SIAM J. NUMER. ANAL. c 2009 Society for Industrial and Applied Mathematics
Vol. 47, No. 2, pp. 997­1018
ACCELERATED LINE-SEARCH AND TRUST-REGION METHODS
P.-A. ABSIL AND K. A. GALLIVAN
Abstract. In numerical optimization, line-search and trust-region methods are two important
classes of descent schemes, with well-understood global convergence properties. We say that these
methods are "accelerated" when the conventional iterate is replaced by any point that produces at
least as much of a decrease in the cost function as a fixed fraction of the decrease produced by the
conventional iterate. A detailed convergence analysis reveals that global convergence properties of
line-search and trust-region methods still hold when the methods are accelerated. The analysis is
performed in the general context of optimization on manifolds, of which optimization in Rn is a
particular case. This general convergence analysis sheds new light on the behavior of several existing
algorithms.
Key words. line search, trust region, subspace acceleration, sequential subspace method, Rie-
mannian manifold, optimization on manifolds, Riemannian optimization, Arnoldi, Jacobi­Davidson,
locally optimal block preconditioned conjugate gradient (LOBPCG)
AMS subject classifications. 65B99, 65K05, 65J05, 65F15, 90C30
DOI. 10.1137/08072019X
1. Introduction. Let f be a real-valued function defined on a domain M, and
let {xk} be a sequence of iterates generated as follows: for every k, some xk+1/2 M

  

Source: Absil, Pierre-Antoine - Département d'ingénierie Mathématique, Université Catholique de Louvain

 

Collections: Mathematics