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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 standard texts, e.g., section 18.1 of \cite{Nocedal2006}.  A similar approach for phase diagram calculation was reported by Lukas \cite{Lukas1982}.  \[ F(T, P, f^{i}, y^i_{s, j}) = \sum_i f^{i}G^{i}_{m}(T, P, y^i_{s, j})\]  \[  L(T, P, f^{i}, y^i_{s, j}, \lambda_k) = \sum_i f^{i}G^{i}_{m}(T, F(T,  P, f^{i},  y^i_{s, j}) - \sum_k \lambda_k c_k(T, P, f^{i}, y^i_{s, j})\] \[ \left[ \begin{array}{ccc}  W_k & -A^T_k \\\  A_k & 0 \end{array} \right]\left[\begin{array}{ccc}  p_k \\\  p_\lambda \end{array} \right]\left[\begin{array}{ccc}  -\nabla f_k F_k  + A^T_k \lambda_k \\\ -c_k \end{array} \right]\]  \(p\) is the Newton step direction for the next iteration. \(W_k\) is the Hessian matrix of \(L\) with respect to all degrees of freedom except \(\lambda_k\). \(A_k\) is the Jacobian matrix of the constraints, \([\nabla c_1, \nabla c_2, ..., \nabla c_m]^T\).  The key advantage of this approach is you can optimize the degrees of freedom of all phases simultaneously.