A New Global Optimization Algorithm and Its Application to Phase Stability Problems

Finding all solutions of mathematical models is still a challenging problem in many areas of science and engineering. There are various methods for finding one or more solutions of the models but they do not guarantee that the solution is a global one or that all solutions have been found, which can be of paramount importance for some modeling problems of physical systems. E.g. in liquid phase stability analysis, we have to find all stationary points of a tangent plane to the Gibbs free energy surface to be sure whether the examined liquid is stable or not. 

In this paper, we describe a general approach to finding all singular points of mathematical models by exploring the natural connectedness that exists between singular points of an objective function. The idea of following ridges and valleys using information gathered along the way is not new and was used in many of the hill climbing or direct search optimization techniques developed in the 1960s. However, these optimization techniques do not use derivative information to determine the search direction, have difficulties tracking strong curvature and are not useful for large-scale problems. They are generally considered to be less efficient than derivative-based methods. 

Later, Van Dongen and Doherty [Chem. Eng. Sci., 1984] used ideas from differential geometry and the theory of differential equations to describe dynamics of distillation processes. They defined a ridge on a topographic map that contains contour lines for different elevations as a location where the contour lines exhibit maximum curvature. They gave also another definition for ridges (valleys) as maxima (minima) of an objective function F(x) taken in the constraint plane normal to the ridge-top locus. The second definition, together with some properties of the gradient system, was used by Wasylkiewicz et al. [Ind. Eng. Chem. Res., 1996] to develop a robust global algorithm for implementation of the Gibbs tangent plane analysis for complex liquid mixtures that involves locating all the stationary points of the tangent plane distance function by tracking its ridges and valleys. They conjectured that the ridges and valleys of F(x) are all connected to each other and that this connected domain contains all the stationary points. It was proved later by Cheng-Dong and Xiao-Xia [Chaos, Solitons and Fractals, 2001]. 

Valleys and ridges of the least-squares landscape are specific integral curves of the gradient vector field and have been defined by Lucia and Feng [AIChE J., 2003] as a collection of constrained extrema over a set of level curves. Based on this definition they developed a so called “global terrain method”, which consists of a series of downhill, equation-solving computations and uphill, predictor–corrector calculations. They noticed that the nonlinear optimization problems used to define valleys and ridges can be quite difficult to solve because they are inherently poorly scaled. 

To overcome these difficulties with efficient tracking of the ridges and valleys of an objective function in the global optimization algorithm we have applied a modified homotopy method together with an arc length continuation originally developed by Wasylkiewicz et al. [Ind. Eng. Chem. Res., 1999] to find all homogeneous as well as heterogeneous azeotropes predicted by a thermodynamic model. First we created the Lagrangian function for the optimization problem and applied Kuhn-Tucker conditions. Then we added an additional equation defining the arc length to the system of original equations to form an augmented system. During branch tracking we look for bifurcation points, from which additional branches are later started in eigendirections. 

As an example, we apply this new global optimization algorithm to the Gibbs tangent plane stability test for multiphase liquid mixtures. The algorithm relies on a new homotopy method that together with arc length continuation gives an efficient and robust scheme for locating all stationary points of the tangent plane distance function predicted by a thermodynamic model. The algorithm is self-starting and significantly improves reliability and robustness of multiphase equilibrium calculations.