Abstract

In this paper, we consider a class of optimization problems with orthogonality constraints, the feasible region of which is called the Stiefel manifold. Our new framework combines a function value reduction step with a correction step. Different from the existing approaches, the function value reduction step of our algorithmic framework searches along the standard Euclidean descent directions instead of the vectors in the tangent space of the Stiefel manifold, and the correction step further reduces the function value and guarantees a symmetric dual variable at the same time. We construct two types of algorithms based on this new framework. The first type is based on gradient reduction including the gradient reflection (GR) and the gradient projection (GP) algorithms. The other one adopts a columnwise block coordinate descent (CBCD) scheme with a novel idea for solving the corresponding CBCD subproblem inexactly. We prove that both GR/GP with a fixed step size and CBCD belong to our algorithmic framework, and any clustering point of the iterates generated by the proposed framework is a first-order stationary point. Preliminary experiments illustrate that our new framework is of great potential.

Keywords

  1. orthogonality constraint
  2. Stiefel manifold
  3. Householder transformation
  4. gradient projection
  5. block coordinate descent

MSC codes

  1. 15A18
  2. 65F15
  3. 65K05
  4. 90C06

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Information & Authors

Information

Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 302 - 332
ISSN (online): 1095-7189

History

Submitted: 13 October 2016
Accepted: 6 October 2017
Published online: 1 February 2018

Keywords

  1. orthogonality constraint
  2. Stiefel manifold
  3. Householder transformation
  4. gradient projection
  5. block coordinate descent

MSC codes

  1. 15A18
  2. 65F15
  3. 65K05
  4. 90C06

Authors

Affiliations

Funding Information

Hong Kong Research Council : N PolyU504/14
Chinese Academy of Sciences https://doi.org/10.13039/501100002367
Chinese Academy of Sciences Key Project https://doi.org/10.13039/501100005151
National Natural Science Foundation of China https://doi.org/10.13039/501100001809 : 11471325, 91530204, 11622112, 11331012, 11461161005

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