Kinetic equations are used to model many physical phenomena, including gas

dynamics, semiconductors, radiative transport, and more. However, high

dimensionality of the domain of definition of the system makes simulation

difficult. The entropy-based moment closure model of the kinetic equation

reduces the dimension of the domain and has attractive theoretical and practical

properties, but most implementations have avoided numerically solving the

defining optimization problem. We use the linear one-dimensional slab-geometry

model to expose the main challenges in the use of numerical optimization then

propose an isotropic regularization and describe the benefits of using fixed

quadrature. A numerical technique using adaptive polynomial bases

in the optimization algorithm is also tested. We develop manufactured

solutions to test our algorithm and also present its performance on two standard

test problems.