A Constraint Dissolving Approach for Nonsmooth Optimization over the Stiefel Manifold

TL;DR

This paper introduces a novel constraint dissolving function for nonsmooth optimization on the Stiefel manifold, enabling the direct application of unconstrained methods and improving efficiency in solving such problems.

Contribution

The paper proposes a new constraint dissolving function, NCDF, that preserves stationary points and simplifies subdifferential computation, facilitating unconstrained optimization methods on the Stiefel manifold.

Findings

NCDF preserves first-order stationary points and local minima.

The Clarke subdifferential of NCDF is easily derived from the objective.

Numerical experiments show improved efficiency of the proposed approach.

Abstract

This paper focus on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same first-order stationary points and local minimizers as the original problem in a neighborhood of the Stiefel manifold. Furthermore, we show that the Clarke subdifferential of NCDF is easy to achieve from the Clarke subdifferential of the objective function. Therefore, various existing approaches for unconstrained nonsmooth optimization can be directly applied to nonsmooth optimization problems over the Stiefel manifold. We propose a framework for developing subgradient-based methods and establish their convergence properties based on prior works. Furthermore, based on our proposed framework, we can develop efficient approaches for optimization over the Stiefel manifold. Preliminary numerical experiments further highlight that the proposed constraint dissolving approach yields efficient and direct implementations of various unconstrained approaches to nonsmooth optimization problems over the Stiefel manifold.