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## Alternative Formulation  The refinement sub-problem can also be solved by constructing an augmented Hessian matrix including all phases and constraints. constraints \cite{Nocedal2006}.  This is similar to the approach of Lukas \cite{Lukas1982}.  The advantage of this approach is you can solve for the chemical potentials and site fractions of all phases simultaneously.  The disadvantages of this are most apparent in large multi-component systems and systems with miscibility gaps;  the Hessian matrix will either  be sparse large  and sparse or it  will have to be rebuilt every time the stable set of phases changes. \[ \chi(\lambda) \textrm{min }L  = \left| \begin{array}  \lambda \sum_i f^{i}G^{i}_{m}(T, P, y^i_{s, j})  - a & -b & -c \\  -d & \lambda - e & -f \\  -g \sum_k \lambda_k c_k(T, P, f^{i}, y^i_{s, j})\]  \[f(x) = \left\{  \begin{array}{lr}  x^2  & -h : x < 0\\  x^3  & \lambda - i : x \ge 0  \end{array} \right|.\] \right.  \]