The HELD (Helmholtz free Energy Lagrangian Dual) algorithm is proposed to solve the isothermal, isobaric phase equilibrium problem (P, T flash). The flash is posed as a minimisation of the Helmholtz free energy in the volume-composition space, reformulated through duality theory. The proposed solution strategy consists of: an initialisation stage, containing a stability test; a phase identification stage, in which linear and nonconvex optimisation problems are solved alternatively; and an acceleration and convergence stage. The stability test is solved with a tunneling algorithm and the nonconvex part of the second stage with a multistart approach. Examples are presented for three equations of state, SRK, SAFT-HS and SAFT-VR. Non-ideal mixtures of up to fifteen components are examined; they exhibit features such as azeotropy, liquid-liquid, and liquid-liquid-liquid equilibria. The HELD algorithm is found to be reliable over a variety of challenging phase behaviour, converging to the best known solution in all of the calculations undertaken. (C) 2011 Elsevier Ltd. All rights reserved.