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A method to utilize the Simplex method and Duality theory of linear programming to solve the nonlinear problem of complex chemical equilibrium calculations of nonideal multiphase reacting systems is presented. The Simplex method and Duality theory are used to automatically generate an optimal set of initial estimates of equilibrium compositions and Lagrangian multipliers to overcome the divergence problem encountered in the iterative solution process. Such an optimization technique has been incorporated into a general purpose equilibrium computational code applicable for ideal or nonideal reacting mixtures that may contain single-phase or multiphase, electrolytic or nonelectrolytic solutions, or miscible or immiscible liquid solutions. Predictions are found in good agreement with experimental data of the binary and ternary solubilities of lithium-lithium fluoride-lithium sulfide system. Complete equilibrium …