Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Simple Algorithms for Optimization on Riemannian Manifolds with Constraints

Simple Algorithms for Optimization on Riemannian Manifolds with Constraints We consider optimization problems on manifolds with equality and inequality con- straints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the Euclidean case to the Riemannian case. Thus, the variable lives on a known smooth manifold and is further constrained. In doing so, we exploit the growing literature on unconstrained Riemannian optimization. For the special case where the manifold is itself described by equality constraints, one could in principle treat the whole problem as a con- strained problem in a Euclidean space. The main hypothesis we test here is whether it is sometimes better to exploit the geometry of the constraints, even if only for a subset of them. Specifically, this paper extends an augmented Lagrangian method and smoothed versions of an exact penalty method to the Riemannian case, together with some fundamental convergence results. Numerical experiments indicate some gains in computational efficiency and accuracy in some regimes for minimum balanced cut, non-negative PCA and k-means, especially in high dimensions. Keywords Riemannian optimization · Constrained optimization · Differential geometry · Augmented Lagrangian method · Exact penalty method · Nonsmooth optimization Mathematics Subject Classification 65K05 · 90C30 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Simple Algorithms for Optimization on Riemannian Manifolds with Constraints

Loading next page...
 
/lp/springer-journals/simple-algorithms-for-optimization-on-riemannian-manifolds-with-t211QRk0va

References (60)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-019-09564-3
Publisher site
See Article on Publisher Site

Abstract

We consider optimization problems on manifolds with equality and inequality con- straints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the Euclidean case to the Riemannian case. Thus, the variable lives on a known smooth manifold and is further constrained. In doing so, we exploit the growing literature on unconstrained Riemannian optimization. For the special case where the manifold is itself described by equality constraints, one could in principle treat the whole problem as a con- strained problem in a Euclidean space. The main hypothesis we test here is whether it is sometimes better to exploit the geometry of the constraints, even if only for a subset of them. Specifically, this paper extends an augmented Lagrangian method and smoothed versions of an exact penalty method to the Riemannian case, together with some fundamental convergence results. Numerical experiments indicate some gains in computational efficiency and accuracy in some regimes for minimum balanced cut, non-negative PCA and k-means, especially in high dimensions. Keywords Riemannian optimization · Constrained optimization · Differential geometry · Augmented Lagrangian method · Exact penalty method · Nonsmooth optimization Mathematics Subject Classification 65K05 · 90C30

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Mar 28, 2019

There are no references for this article.