# GRADIENT PROJECTION METHODS FOR THE n-COUPLING PROBLEM

• Kum, Sangho (Department of Mathematics Education Chungbuk National University) ;
• Yun, Sangwoon (Department of Mathematics Education Sungkyunkwan University)
• Accepted : 2018.08.29
• Published : 2019.07.01
• 103 10

#### 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

#### File

FIGURE 1. (a) Objective value versus iteration with eiglb = 1, eigub = 99. (b) Objective value versus iteration with eiglb = 0:1, eigub = 99:9. The estimated Lipschitz constant in (b) is more than 10 times bigger than that in (a). It is observed that the estimated Lipschitz constant is smaller, the gap between the objective value of AGPM and that of GPM-A is bigger.

TABLE 1. Test results of the ﬁnal objective values and CPU time in seconds for three methods GPM-A, GPM-C, and AGPM on 5 random data sets.

#### Acknowledgement

Supported by : NRF

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