This paper tackles the problem of computing the Karcher mean of a collection of symmetric positive-definite matrices. We present a concrete limited-memory Riemannian BFGS method to handle this computational task. We also provide methods to produce efficient numerical representations of geometric objects on the manifold of symmetric positive-definite matrices that are required for Riemannian opt...

A new method for solving large nonlinear optimization problems is outlined. It attempts to combine the best properties of the discrete-truncated Newton method and the limited memory BFGS method, to produce an algorithm that is both economical and capable of handling ill-conditioned problems. The key idea is to use the curvature information generated during the computation of the discrete Newton...

In theory, the successive gradients generated by the conjugate gradient method applied to a quadratic should be orthogonal. However, for some ill-conditioned problems, orthogonality is quickly lost due to rounding errors, and convergence is much slower than expected. A limited memory version of the nonlinear conjugate gradient method is developed. The memory is used to both detect the loss of o...

This study concerns with a trust-region-based method for solving unconstrained optimization problems. The approach takes the advantages of the compact limited memory BFGS updating formula together with an appropriate adaptive radius strategy. In our approach, the adaptive technique leads us to decrease the number of subproblems solving, while utilizing the structure of limited memory quasi-Newt...

The focus of this dissertation is on matrix decompositions that use a limited amount of computer memory, thereby allowing problems with a very large number of variables to be solved. Speciically, we will focus on two applications areas: optimization and information retrieval. We introduce a general algebraic form for the matrix update in limited-memory quasi-Newton methods. Many well-known meth...

The focus of this dissertation is on matrix decompositions that use a limited amount of computer memory, thereby allowing problems with a very large number of variables to be solved. Speciically, we will focus on two applications areas: optimization and information retrieval. We introduce a general algebraic form for the matrix update in limited-memory quasi-Newton methods. Many well-known meth...