Test Problem

The test problem used is a global nonlinear parameter estimation problem involving a vapor-liquid equilibrium (VLE) model (Wilson's equation). Such models, and the estimation of parameters in them, are important in chemical process engineering, since they are the basis for the design, simulation and optimization of widely-used separation processes such as distillation [48]. In this particular problem, we use as the objective function the maximum likelihood estimator, with two unknown standard deviations, to determine two model parameters giving the globally optimal fit of the data to the model [84]. In addition to the difficult nonlinear objective function, the problem data and characteristics were chosen to make this a particularly difficult problem, requiring a few hours of computation time on a single processor. Interval analysis, as described above, is used to guarantee the correct global solution. The problem can be solved in either of two ways. One approach is to treat it as a nonlinear equation solving problem, and use the parallel interval BP algorithm to solve for all stationary points of the objective function (there are five stationary points in this problem). The alternative approach is to treat it directly as a global optimization problem and use the parallel interval BB algorithm. The major difference between the two approaches is the use of the objective range test in the BB algorithm.