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 Gradient projection methods for the $n$-coupling problem J. Korean Math. Soc. 2019 Vol. 56, No. 4, 1001-1016 https://doi.org/10.4134/JKMS.j180517Published online July 1, 2019 Sangho Kum, Sangwoon Yun Chungbuk National University; Sungkyunkwan University Abstract : We are concerned with optimization methods for the $L^2$-Wasserstein least squares problem of Gaussian measures (alternatively the $n$-coupling problem). Based on its equivalent form on the convex cone of positive definite matrices of fixed size and the strict convexity of the variance function, we are able to present an implementable (accelerated) gradient method for finding the unique minimizer. Its global convergence rate analysis is provided according to the derived upper bound of Lipschitz constants of the gradient function. Keywords : $L^2$-Wasserstein least squares problem, $n$-coupling problem, Gaussian measure, positive definite matrix, Nesterov-Todd scaling, gradient projection method MSC numbers : Primary 49M30, 90C25 Full-Text :