Multipliers Correction Methods for Optimization Problems over the Stiefel Manifold

TL;DR

This paper introduces multipliers correction methods for optimization over the Stiefel manifold, combining descent and proximal steps, with proven convergence and superior performance in tests.

Contribution

It presents a novel class of methods that improve upon existing approaches by integrating Euclidean descent with proximal corrections for better optimization over the Stiefel manifold.

Findings

Methods outperform state-of-the-art approaches in numerical tests.

Global convergence of the proposed methods is established.

Significant reduction in objective function observed in experiments.

Abstract

We propose a class of multipliers correction methods to minimize a differentiable function over the Stiefel manifold. The proposed methods combine a function value reduction step with a proximal correction step. The former one searches along an arbitrary descent direction in the Euclidean space instead of a vector in the tangent space of the Stiefel manifold. Meanwhile, the latter one minimizes a first-order proximal approximation of the objective function in the range space of the current iterate to make Lagrangian multipliers associated with orthogonality constraints symmetric at any accumulation point. The global convergence has been established for the proposed methods. Preliminary numerical experiments demonstrate that the new methods significantly outperform other state-of-the-art first-order approaches in solving various kinds of testing problems.