The present paper describes the coupling of the Mesh Adaptive Direct Search (MADS) algorithm with the FactSage thermochemical software, which allows to calculate the global and local minima of the liquidus temperature for a given multicomponent system provided that an accurate thermodynamic database has been previously developed for this system. There is actually no limit on the number of components. As examples, the global minimum and local minima of the liquidus temperature are calculated for the MgO-CaO-Al₂O₃-SiO₂ oxide system, the LiCl-NaCl-KCl-MgCl₂-CaCl₂-SrCl₂-BaCl₂ salt system and the Fe-C-Co-Cr-Mn-Mo-Si-V-W metallic system. In particular, the global minima calculated for the Fe-C-Cr-Mo-V and Fe-C-Cr-Mn-Mo subsystems of the latter system compare very well with the results previously obtained by other groups using the Thermo-Calc software along with the SGTE database.