We present recent progress on the implementation of entropy-based moment closures in the context of a simple, linear kinetic equation. The algorithm has two main, coupled components: a second-­order kinetic scheme to update the PDE and a Newton-­based solver for the dual of the optimization problem that defines the closure. We study in detail the difficulties of solving the dual problem near the boundary of realizable moments, where quadrature formulas are less reliable and the Hessian of the dual objective function is highly ill-­?conditioned. Extensive numerical experiments are performed to illustrate these difficulties. In cases where the dual problem becomes "too difficult" to solve numerically, we propose a regularization technique to artificially move moments away from the realizable boundary in a way that still preserves local particle concentrations. Results are given for benchmark problems in one and two dimensions. In the latter case, a strategy for parallelization on heterogeneous architectures has been devised in order to reduce the high cost of solving millions of optimization problems. This is joint work with Graham Alldredge (RWTH Aachen), Kris Garrett (Oak Ridge), and Dianne O'Leary (U. Maryland) and Andre Tits (U. Maryland).