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Optimization Letters

, Volume 13, Issue 8, pp 1773–1791 | Cite as

First order optimality conditions and steepest descent algorithm on orthogonal Stiefel manifolds

  • Petre BirteaEmail author
  • Ioan Caşu
  • Dan Comănescu
Original Paper

Abstract

Considering orthogonal Stiefel manifolds as constraint manifolds, we give an explicit description of a set of local coordinates that also generate a basis for the tangent space in any point of the orthogonal Stiefel manifolds. We show how this construction depends on the choice of a submatrix of full rank. Embedding a gradient vector field on an orthogonal Stiefel manifold in the ambient space, we give explicit necessary and sufficient conditions for a critical point of a cost function defined on such manifolds. We explicitly describe the steepest descent algorithm on the orthogonal Stiefel manifold using the ambient coordinates and not the local coordinates of the manifold. We point out the dependence of the recurrence sequence that defines the algorithm on the choice of a full rank submatrix. We illustrate the algorithm in the case of Brockett cost functions.

Keywords

Steepest descent algorithm Optimization Constraint manifold Orthogonal Stiefel manifold Brockett cost function 

Notes

Acknowledgements

This work was supported by a Grant of Ministery of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2016-0165, within PNCDI III.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsWest University of TimişoaraTimişoaraRomania

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