Conditional Modeling. 2. Solving Using Complementarity and Boundary-Crossing Formulations

Abstract

In this second paper on conditional modeling, we investigate the solving of the special subclass of conditional models which display continuous values for the problem variables as one crosses from one region to the next for such models. First, we present a new complementarity formulation for representing algebraic systems of disjunctive equations, and then we discuss and demonstrate the use of a boundary-crossing algorithm proposed by Zaher (Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 1995). Our complementarity formulation not only establishes the complementarity condition among equations belonging to different disjunctive terms but also enforces simultaneous satisfaction of all the equations appearing within the same disjunctive term. This approach performed reliably on several example problems where the number of equations in each disjunctive term is small. Zaher also proposed the boundary-crossing algorithm as an alternative to using MINLP techniques for solving the special subclass of continuous conditional models. This algorithm involves the execution of several well-differentiated activities including logical analysis and continuous reconfiguration of the equations constituting the problem, calculation of Newton-like steps, and calculation of subgradient steps. We describe the practical implementation of the boundary-crossing algorithm as a conditional-modeling solution tool. In such an implementation, we have integrated the entities created and/or used to perform each of the activities in an object-oriented solving engine: the conditional-modeling solver CMSlv. We solve several examples of conditional models in chemical engineering by using both the approaches for solving, and we identify the advantages and disadvantages of such approaches.