Solving Optimization Problems over the Stiefel Manifold by Smooth Exact Penalty Functions (Journal of Computational Mathematics, 2024)
In this paper, we present a novel penalty model called ExPen for optimization over theStiefel manifold. Different from existing penalty functions for orthogonality constraints,ExPen adopts a smooth penalty function without using any first-order derivative of theobjective 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 stationarypoints of the original optimization problem, or far from the Stiefel manifold. Besides, theoriginal 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 straightforwardlyto solve ExPen.