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 - <p style="text-align: justify;">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.</p>