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We introduce the $\ell_0\ell_2$-norm regularization and hierarchy constraintsinto linear regression for the construction of cluster expansion to describeconfigurational disorder in materials. The approach is implemented throughmixed integer quadratic programming (MIQP). The $\ell_2$-norm regularization isused to suppress intrinsic data noise, while $\ell_0$-norm is used to penalizethe number of non-zero elements in the solution. The hierarchy relation betweenclusters imposes relevant physics and is naturally included by the MIQPparadigm. As such, sparseness and cluster hierarchy can be well optimized toobtain a robust, converged, and effective cluster interactions with improvedphysical meaning. We demonstrate the effectiveness of $\ell_0\ell_2$-normregularization in two high-component disordered rocksalt cathode materialsystems, where we compare the cross-validation and convergence speed,reproduction of phase diagrams, voltage profiles, and Li-occupancy energieswith those of the conventional $\ell_1$-norm regularized cluster expansionmodel.

An $\ell_0\ell_2$-norm regularized regression model for construction of robust cluster expansions in multicomponent systems