  We introduce the $\ell_0\ell_2$-norm regularization and hierarchy constraints
into linear regression for the construction of cluster expansion to describe
configurational disorder in materials. The approach is implemented through
mixed integer quadratic programming (MIQP). The $\ell_2$-norm regularization is
used to suppress intrinsic data noise, while $\ell_0$-norm is used to penalize
the number of non-zero elements in the solution. The hierarchy relation between
clusters imposes relevant physics and is naturally included by the MIQP
paradigm. As such, sparseness and cluster hierarchy can be well optimized to
obtain a robust, converged, and effective cluster interactions with improved
physical meaning. We demonstrate the effectiveness of $\ell_0\ell_2$-norm
regularization in two high-component disordered rocksalt cathode material
systems, where we compare the cross-validation and convergence speed,
reproduction of phase diagrams, voltage profiles, and Li-occupancy energies
with those of the conventional $\ell_1$-norm regularized cluster expansion
model.
