Moment equations are an attractive approach to solving the Boltzmann equation for rarefied gas dynamics. However, to obtain a closed system of partial differential equations for N moments of the velocity distribution function (VDF), knowledge of N+1 moments is required. In the general case, an analytical form of the VDF is not known beforehand, and thus, some numerical approach is required to obtain the higher-order moments (usually via reconstruction of the VDF and subsequent computation of the relevant moments). These include entropy-based approaches [Levermore 1996] and quadrature methods [Fox 2008]. In general, the closure problem can be viewed as a constrained optimization problem: given a set of basis functions in velocity space and known moments, find the weights for a weighted sum of the basis functions such that it reproduces the known moments, minimizes some prescribed target functional (e.g., entropy), and possesses certain properties (for example, non-negativity). In the present work, different target functionals are considered, such as the sparsity-promoting L_1 norm, the regularizing L_2 norm, entropy, as well as combinations thereof. The approaches are tested on various families of gas-dynamics distributions, as well as synthetic distributions. Issues with realizability and algorithm robustness are investigated, along with possible mitigation strategies [Alldredge 2019, Sadr 2023].