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

Nachuan Xiao; Xin Liu

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arXiv · 2110.08986

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

Nachuan Xiao·Xin Liu

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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.

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BibTeX RIS

Nachuan Xiao, Xin Liu. 2021-10-18. Solving Optimization Problems over the Stiefel Manifold by Smooth Exact Penalty Function. https://arxiv.org/abs/2110.08986

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