ITERATIVE LINEAR-PROGRAMMING STRATEGIES FOR CONSTRAINED SIMULATION

With the development of powerful equation-oriented process simulators, such as SPEEDUP and ASCEND, much research deals with the development of reliable and efficient nonlinear equation solvers for process simulation. However, these methods often approach the treatment of variable bounds or other inequality constraints with ad hoc strategies. Moreover, the absence of a suitable systematic strategy can lead to convergence failures. Instead, we propose an iterative linear programming (LP) strategy which automatically deals with inequality constraints and variable bounds. This method reduces to Newton's method in the absence of inequalities and thus has the ability to converge quadratically in the neighborhood of the solution. Moreover, this LP approach can be shown to generate a descent direction for the 1-norm of the constraint violations. Therefore, by applying a line search in the algorithm, one can guarantee progress toward the solution for each nonzero solution to the LP. Other advantages to this approach are that it can deal with many classes of singular points, and the path toward the solution is always contained within the variable bounds. This approach performs very well on a number of small problems and has been implemented within a large-scale modelling system (GAMS). This allows an integration of the method with powerful large-scale LP solvers as well as facilities for quickly generating large process engineering problems. Here we will demonstrate this algorithm for the solution of example problems including pipeline networks and ideal and nonideal distillation problems. It should be mentioned, however, that this algorithm can fail when the LP yields a zero search direction at a nonsolution point. We term this special class of stationary points "pseudosolutions" and derive theoretical properties which characterize them. To recover from pseudosolutions, an "epsilon-strategy" is applied which shifts an active constraint into the feasible region until an improved point is found. While this approach is heuristic, it was effective in restarting and successfully solving all of the cases we encountered.