DIRECT SEARCH ALGORITHMS OVER RIEMANNIAN MANIFOLDS

David W. Dreisigmeyer (dreisigmlanl.gov)

Abstract: We generalize the Nelder-Mead simplex and LTMADS algorithms and, the frame based methods for function minimization to Riemannian manifolds. Examples are given for functions defined on the special orthogonal Lie group $\mathcal{SO}(n)$ and the Grassmann manifold $\mathcal{G}(n,k)$. Our main examples are applying the generalized LTMADS algorithm to equality constrained optimization problems and, to the Whitney embedding problem for dimensionality reduction of data. A convergence analysis of the frame based method is also given.

Keywords: Direct search methods, Riemannian manifolds, Grassmann manifolds, equality constraints, geodesics

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

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Entry Submitted: 08/02/2007
Entry Accepted: 08/02/2007
Entry Last Modified: 08/02/2007

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