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# Introduction  Miscibility gap detection is a crucial feature in thermodynamic calculation software to accurately calculate the energy of phases containing regions of compositional instability, i.e., spinodals, and is commonly handled through global minimization (GM) of the Gibbs energy. Inside the spinodal region, all points on the energy surface have negative curvature and will be thermodynamically driven towards demixing. The cause of miscibility gaps in non-ideal solutions is the presence of unfavorable interactions between components that overwhelm the entropically-driven ideal mixing contribution to the Gibbs energy. From a computational perspective miscibility gaps pose a challenge because they mean that the same phase may have multiple points on the equilibrium tie hyperplane (tie plane in ternary systems), but at different compositions. In these cases the software must increase the total degrees of freedom by creating multiple composition sets of the same phase, potentially up to the limit specified by the Gibbs phase rule. For multi-component systems the topology of the energy surfaces can become quite complex. Moreover, when handling phases with internal degrees of freedom, i.e., sublattices, it is possible for points on the global energy surface (the composition) to be close together while being far apart in their internal coordinates (the constitution). This is referred to as an internal miscibility gap or second-order phase transition. While few authors have discussed fully generalized GM schemes, there has been work in efficient sampling in low dimensions \cite{Emelianenko_2006} and tie hyperplane calculation for the multi-component case \cite{Perevoshchikova_2012}. However, there has not previously been detailed discussion of methods for solving the multi-component case with multiple sublattices.