0 ≤ x i n ≤ 1, for i ∈ [0 . . l] and n ∈ [0 . . z] (5) 0 < r i ≤ 1, for i ∈ [0 . . l] (6) All dopant concentrations are within the closed interval [0, 1]. The fractional radii is defined on an interval of [0, 1] as a fraction of a the maximum nanoparticle size, r max (which we identify a priori). This is necessary to keep on the same interval as the concentration, since the trust region optimizer defines the same trust region for all independent variables r f raction = r true /r max In addition, we define linear constraints that bound the total concentration within each layer [0, 1] and restrict the thickness of each layer.