Wasserstein barycenter is the centroid of a collection of discrete probability distributions which minimizes the average of the 2 -Wasserstein distance. This paper focuses on the computation of Wasserstein barycenters under the case where the support points are free, which is known to be a severe bottleneck in the D2-clustering due to the large scale and nonconvexity. We develop an inexact proximal alternating minimization (iPAM) method for computing an approximate Wasserstein barycenter, and provide its global convergence analysis. This method can achieve a good accuracy with a reduced computational cost when the unknown support points of the barycenter have low cardinality. Numerical comparisons with the 3-block B-ADMM in Ye et al. (IEEE Trans Signal Process 65:2317-2332) and an alternating minimization method involving the LP subproblems on synthetic and real data show that the proposed iPAM can yield comparable even a little better objective values in less CPU time, and hence, the computed barycenter will render a better role in the D2-clustering.

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Wasserstein barycenter is the centroid of a collection of discrete probability distributions which minimizes the average of the ℓ 2 -Wasserstein distance. This paper focuses on the computation of Wasserstein barycenters under the case where the support points are free, which is known to be a severe bottleneck in the D2-clustering due to the large-scale and nonconvexity. We develop an inexact proximal alternating minimization (PAM) method for computing an approximate Wasserstein barycenter, and provide its global convergence analysis. This method can achieve a good accuracy with a reduced computational cost when the unknown support points of the barycenter have low cardinality. Numerical comparisons with the 3-block B-ADMM proposed in [30] on synthetic and real data indicate that our method yields comparable even a little better objective values within less computing time, and hence the computed approximate barycenter will render a better role in the D2-clustering.

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.