Normal alkanes show very complicated phase transition kinetics and macroscopic phase equilibrium behavior. This paper focuses on the phase stability and equilibrium of complicated mixtures like n-alkanes and on the enabling global optimization technologies needed to gather problem knowledge. The new ideas contained in this paper include: (1) novel level set methods for gathering encoded knowledge; (2) the differential geometry for uncovering pathways to more subtle knowledge; (3) all supporting non-linearly constrained optimization techniques; (4) all data handling needed to unravel complex solution structure. These new ideas are incorporated within the integral path methodology or terrain methods recently developed by Lucia, A., & Yang, F. (2003) [Lucia, A., & Yang F. (2003). Multivariable terrain methods. AIChE Journal 49, 2553] and Lucia, DiMaggio, and Depa (2004) [Lucia, A., DiMaggio, P.A., & Depa, P. (2004). A geometric terrain methodology for global optimization. Journal of Global Optimization 29, 297]. This framework provides global knowledge for understanding solution structure, like the complex solution structure of n-alkanes. In particular, it is shown that knowledge of the Newton and tangent vector fields, Gauss curvature, integral path bifurcation points, and non-differentiable manifolds provides a deterministic way of finding additional solutions, saddle points, and other information that might otherwise go undetected. It is shown that the optimization tools developed in this work provide all knowledge of interest on the appropriate hypothetical single-phase or composite surface (i.e., minima, saddle points, singular points, and integral paths) in phase stability applications. This knowledge can be obtained by solving the phase stability problem exactly once, in a pre-processing step, and used to reliably initialize any multi-phase equilibrium calculation for any feed. This removes the need to repeatedly solve the phase stability problem as the feed composition changes and greatly increases computational efficiency. Numerical examples and geometric illustrations are used to elucidate key ideas and to show how the proposed approach can be used to unravel the complicated phase behavior of n-alkane mixtures. (c) 2005 Elsevier Ltd. All rights reserved.