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

Abstract

Abstract This paper is concerned with a class of optimization problems with the non-negative orthogonal constraint, in which the objective function is $L$-smooth on an open set containing the Stiefel manifold $\textrm {St}(n,r)$. 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 non-negative 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 [Wen, Z. W. & Yin, W. T. (2013, A feasible method for optimization with orthogonality constraints. Math. Programming, 142, 397–434),] and the exact penalty method [Jiang, B., Meng, X., Wen, Z. W. & Chen, X. J. (2022, An exact penalty approach for optimization with nonnegative orthogonality constraints. Math. Programming. https://doi.org/10.1007/s10107-022-01794-8)] indicate that our penalty method has an advantage in terms of the quality of solutions despite taking a little more time.