Entropy-based (M_N) moment closures for kinetic equations are defined by a

constrained optimization problem that must be solved at every point in a

space-time mesh, making it important to solve these optimization problems

accurately and efficiently. We present a complete and practical numerical

algorithm for solving the dual problem in one-dimensional, slab geometries. The

closure is only well-defined on the set of moments that are realizable from a

positive underlying distribution, and as the boundary of the realizable set is

approached, the dual problem becomes increasingly difficult to solve due to

ill-conditioning of the Hessian matrix. To improve the condition number of the

Hessian, we advocate the use of a change of polynomial basis, defined using a

Cholesky factorization of the Hessian, that permits solution of problems nearer

to the boundary of the realizable set. We also advocate a fixed quadrature

scheme, rather than adaptive quadrature, since the latter introduces

unnecessary expense and changes the computationally realizable set as the

quadrature changes. For very ill-conditioned problems, we use regularization to

make the optimization algorithm robust. We design a manufactured solution and

demonstrate that the adaptive-basis optimization algorithm reduces the need for

regularization. This is important since we also show that regularization slows,

and even stalls, convergence of the numerical simulation when refining the

space-time mesh. We also simulate two well-known benchmark problems. There we

find that our adaptive-basis, fixed-quadrature algorithm uses less

regularization than alternatives, although differences in the resulting

numerical simulations are more sensitive to the regularization strategy than to

the choice of basis.