Phase equilibria of multicomponent mixtures are considered and a reinterpretation of the Gibbs tangent plane stability criterion is proposed via Lagrangian duality. The starting point is the natural primal problem of minimizing the Gibbs free energy subject to material balance. The stable phase split is the solution of the corresponding dual problem, providing a necessary and sufficient dual extremum principle. Only in the absence of duality gap is the physical phase split also the solution of the primal problem. The only requirements are continuity of the Gibbs free energy and the trivial requirement that each species is present in the overall composition. The number of phases is permitted to be infinite, and does not need to be known a priori. No assumption is made on the presence of all species in all phases. Case studies are presented based on the NRTL and UNIQUAC activity coefficient model.