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Solving Optimization Problems over the Stiefel Manifold by Smooth Exact Penalty Functions

Solving Optimization Problems over the Stiefel Manifold by Smooth Exact Penalty Functions

Year:    2024

Author:    Nachuan Xiao, Xin Liu

Journal of Computational Mathematics, Vol. 42 (2024), Iss. 5 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.2307-m2021-0331

Journal of Computational Mathematics, Vol. 42 (2024), Iss. 5 : pp. 1246–1276

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    31

Keywords:    Orthogonality constraint Stiefel manifold Penalty function.

Author Details

Nachuan Xiao

Xin Liu