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Volume 42, Issue 5
Solving Optimization Problems over the Stiefel Manifold by Smooth Exact Penalty Functions

Nachuan Xiao & Xin Liu

DOI: 10.4208/jcm.2307-m2021-0331

J. Comp. Math., 42 (2024), pp. 1246-1276.

Published online: 2024-07

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  • Abstract

In this paper, we present a novel penalty model called ExPen for optimization over the Stiefel manifold. Different from existing penalty functions for orthogonality constraints, ExPen adopts a smooth penalty function without using any first-order derivative of the objective function. We show that all the first-order stationary points of ExPen with a sufficiently large penalty parameter are either feasible, namely, are the first-order stationary points of the original optimization problem, or far from the Stiefel manifold. Besides, the original problem and ExPen share the same second-order stationary points. Remarkably, the exact gradient and Hessian of ExPen are easy to compute. As a consequence, abundant algorithm resources in unconstrained optimization can be applied straightforwardly to solve ExPen.

  • Keywords

Orthogonality constraint, Stiefel manifold, Penalty function.

  • AMS Subject Headings

90C30, 65K05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1246, author = {Xiao , Nachuan and Liu , Xin}, title = {Solving Optimization Problems over the Stiefel Manifold by Smooth Exact Penalty Functions}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {5}, pages = {1246--1276}, abstract = {

In this paper, we present a novel penalty model called ExPen for optimization over the Stiefel manifold. Different from existing penalty functions for orthogonality constraints, ExPen adopts a smooth penalty function without using any first-order derivative of the objective function. We show that all the first-order stationary points of ExPen with a sufficiently large penalty parameter are either feasible, namely, are the first-order stationary points of the original optimization problem, or far from the Stiefel manifold. Besides, the original problem and ExPen share the same second-order stationary points. Remarkably, the exact gradient and Hessian of ExPen are easy to compute. As a consequence, abundant algorithm resources in unconstrained optimization can be applied straightforwardly to solve ExPen.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2307-m2021-0331}, url = {http://global-sci.org/intro/article_detail/jcm/23277.html} }
TY - JOUR T1 - Solving Optimization Problems over the Stiefel Manifold by Smooth Exact Penalty Functions AU - Xiao , Nachuan AU - Liu , Xin JO - Journal of Computational Mathematics VL - 5 SP - 1246 EP - 1276 PY - 2024 DA - 2024/07 SN - 42 DO - http://doi.org/10.4208/jcm.2307-m2021-0331 UR - https://global-sci.org/intro/article_detail/jcm/23277.html KW - Orthogonality constraint, Stiefel manifold, Penalty function. AB -

In this paper, we present a novel penalty model called ExPen for optimization over the Stiefel manifold. Different from existing penalty functions for orthogonality constraints, ExPen adopts a smooth penalty function without using any first-order derivative of the objective function. We show that all the first-order stationary points of ExPen with a sufficiently large penalty parameter are either feasible, namely, are the first-order stationary points of the original optimization problem, or far from the Stiefel manifold. Besides, the original problem and ExPen share the same second-order stationary points. Remarkably, the exact gradient and Hessian of ExPen are easy to compute. As a consequence, abundant algorithm resources in unconstrained optimization can be applied straightforwardly to solve ExPen.

Nachuan Xiao & Xin Liu. (2024). Solving Optimization Problems over the Stiefel Manifold by Smooth Exact Penalty Functions. Journal of Computational Mathematics. 42 (5). 1246-1276. doi:10.4208/jcm.2307-m2021-0331
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