An algorithm to directly locate K- or L-points (which are often called critical endpoints) is proposed that utilizes a robust critical point solver and a standard phase stability test within a nested-loop structure. Successive substitution (SS), accelerated successive substitution, and Newton-Raphson (NR) procedures were used to perform the phase stability test, and it was determined that a hybrid between the SS and the NR methods balances the robustness of the SS procedure against the convergence rate of the NR procedure. The algorithm was tested using the Peng-Robinson equation of state for a variety of binary systems, including an ethane + ethanol system, three methane + n-alkane systems, and three propane + polyaromatic systems. An analysis of convergence difficulties is discussed, relative to the Gibbs free-energy surface topology. It was determined that an accurate initial critical composition was necessary to avoid convergence to a trivial solution. Convergence was not sensitive to the initial guess used to locate the noncritical equilibrium phase.