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

Nachuan Xiao (xnc***at***lsec.cc.ac.cn)
Xin Liu (liuxin***at***lsec.cc.ac.cn)

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: Stiefel manifold, Penalty function method, Riemannian optmization

Category 1: Nonlinear Optimization


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Entry Submitted: 10/12/2021
Entry Accepted: 10/12/2021
Entry Last Modified: 11/22/2021

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