Error bound and exact penalty method for optimization problems with nonnegative orthogonal constraint

TL;DR

This paper develops error bounds and an exact penalty method for optimization problems with nonnegative orthogonal constraints, providing theoretical guarantees and a practical algorithm with competitive numerical performance.

Contribution

It introduces new error bounds and proves the global exactness of a penalty approach for nonnegative orthogonal constrained problems, along with a practical solution algorithm.

Findings

Error bounds established for feasible points without zero rows.

The penalty method is proven to be globally exact under certain conditions.

Numerical results show the method produces high-quality solutions.

Abstract

This paper is concerned with a class of optimization problems with the nonnegative orthogonal constraint, in which the objective function is $L$-smooth on an open set containing the Stiefel manifold ${\rm St}(n,r)$. We derive a locally Lipschitzian error bound for the feasible points without zero rows when $n>r>1$, and when $n>r=1$ or $n=r$ achieve a global Lipschitzian error bound. Then, we show that the penalty problem induced by the elementwise $\ell_1$-norm distance to the nonnegative cone is a global exact penalty, and so is the one induced by its Moreau envelope under a lower second-order calmness of the objective function. A practical penalty algorithm is developed by solving approximately a series of smooth penalty problems with a retraction-based nonmonotone line-search proximal gradient method, and any cluster point of the generated sequence is shown to be a stationary point of the original problem. Numerical comparisons with the ALM \citep{Wen13} and the exact penalty method \citep{JiangM22} indicate that our penalty method has an advantage in terms of the quality of solutions despite taking a little more time.