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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
Naman Agarwal, Nicolas Boumal, Brian Bullins, C. Cartis (2018)
Adaptive regularization with cubics on manifolds with a first-order analysis
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
Applied Mathematics and Optimization – Springer Journals
Published: Mar 28, 2019
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