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The refinement sub-problem can also be solved by constructing an augmented Hessian matrix including all phases' degrees of freedom and their constraints.  Iterative steps toward the solution are then computed using the Newton-Raphson method.  This is sometimes called the "Newton-Lagrange" method and is detailed in section 18.1 of \cite{Nocedal2006}.  This is The application of a  similarto the  approach of to phase diagram calculation was reported by  Lukas \cite{Lukas1982}. The key  advantage of this approach is you can optimize the degrees of freedom of all phases simultaneously. The disadvantagesof this  are most apparent in large multi-component systems and systems with miscibility gaps; in principle \(c+2\) copies of *each phase* in the systemwill  need to be entered or else some procedure for adding and removing composition sets is required, in which case the Hessian must be rebuilt every time the stable set of phases changes. \[ L(T, P, f^{i}, y^i_{s, j}, \lambda_k) = \sum_i f^{i}G^{i}_{m}(T, P, y^i_{s, j}) - \sum_k \lambda_k c_k(T, P, f^{i}, y^i_{s, j})\]  However this approach can be useful if one is already close to the solution.  \[ \chi(\lambda) = \left| \begin{array}{ccc}  \lambda - a & -b & -c \\\  -d & \lambda - e & -f \\\  -g & -h & \lambda - i \end{array} \right|.\]