Volume 2, Issue 3
Multipliers Correction Methods for Optimization Problems over the Stiefel Manifold

Lei Wang, Bin Gao & Xin Liu

CSIAM Trans. Appl. Math., 2 (2021), pp. 508-531.

Published online: 2021-08

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  • Abstract

We propose a class of multipliers correction methods to minimize a differentiable function over the Stiefel manifold. The proposed methods combine a function value reduction step with a proximal correction step. The former one searches along an arbitrary descent direction in the Euclidean space instead of a vector in the tangent space of the Stiefel manifold. Meanwhile, the latter one minimizes a first-order proximal approximation of the objective function in the range space of the current iterate to make Lagrangian multipliers associated with orthogonality constraints symmetric at any accumulation point. The global convergence has been established for the proposed methods. Preliminary numerical experiments demonstrate that the new methods significantly outperform other state-of-the-art first-order approaches in solving various kinds of testing problems.

  • Keywords

Multipliers correction, proximal approximation, orthogonality constraint, Stiefel manifold.

  • AMS Subject Headings

15A18, 65F15, 65K05, 90C06, 90C30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-2-508, author = {Lei Wang , and Bin Gao , and Liu , Xin}, title = {Multipliers Correction Methods for Optimization Problems over the Stiefel Manifold}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2021}, volume = {2}, number = {3}, pages = {508--531}, abstract = {

We propose a class of multipliers correction methods to minimize a differentiable function over the Stiefel manifold. The proposed methods combine a function value reduction step with a proximal correction step. The former one searches along an arbitrary descent direction in the Euclidean space instead of a vector in the tangent space of the Stiefel manifold. Meanwhile, the latter one minimizes a first-order proximal approximation of the objective function in the range space of the current iterate to make Lagrangian multipliers associated with orthogonality constraints symmetric at any accumulation point. The global convergence has been established for the proposed methods. Preliminary numerical experiments demonstrate that the new methods significantly outperform other state-of-the-art first-order approaches in solving various kinds of testing problems.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2020-0008}, url = {http://global-sci.org/intro/article_detail/csiam-am/19448.html} }
TY - JOUR T1 - Multipliers Correction Methods for Optimization Problems over the Stiefel Manifold AU - Lei Wang , AU - Bin Gao , AU - Liu , Xin JO - CSIAM Transactions on Applied Mathematics VL - 3 SP - 508 EP - 531 PY - 2021 DA - 2021/08 SN - 2 DO - http://doi.org/10.4208/csiam-am.SO-2020-0008 UR - https://global-sci.org/intro/article_detail/csiam-am/19448.html KW - Multipliers correction, proximal approximation, orthogonality constraint, Stiefel manifold. AB -

We propose a class of multipliers correction methods to minimize a differentiable function over the Stiefel manifold. The proposed methods combine a function value reduction step with a proximal correction step. The former one searches along an arbitrary descent direction in the Euclidean space instead of a vector in the tangent space of the Stiefel manifold. Meanwhile, the latter one minimizes a first-order proximal approximation of the objective function in the range space of the current iterate to make Lagrangian multipliers associated with orthogonality constraints symmetric at any accumulation point. The global convergence has been established for the proposed methods. Preliminary numerical experiments demonstrate that the new methods significantly outperform other state-of-the-art first-order approaches in solving various kinds of testing problems.

Lei Wang, Bin Gao & XinLiu. (2021). Multipliers Correction Methods for Optimization Problems over the Stiefel Manifold. CSIAM Transactions on Applied Mathematics. 2 (3). 508-531. doi:10.4208/csiam-am.SO-2020-0008
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