In this paper we present general formulations of the phase-equilibrium and phase-stability problems for multicomponent mixtures and verify that these formulations generalize the problems of phase equilibrium and phase-stability at constant volume, temperature, and mole numbers (VTN-flash), at constant internal energy, volume, and mole numbers (UVN-flash), and at constant pressure, temperature, and mole numbers (PTN-flash). Furthermore, we develop a numerical method for solving the general formulation of phase-equilibrium problems. This algorithm is based on the direct minimization of the objective function with respect to the constraints. The algorithm uses a modified Newton-Raphson method, along with a modified Cholesky decomposition of the Hessian matrix to generate a sequence of states with decreasing values of the objective function. The algorithm was implemented in C++ and using generic programing we have a single, portable solver for all three flash formulations. Properties of the algorithm are shown on phase-equilibria problems of multicomponent mixtures in different specifications and with different levels of difficulty. Complexities and numerical performance of the individual flash formulations are discussed. (C) 2018 Published by Elsevier B.V.