The use of interval methods, in particular interval-Newton/generalized-bisection techniques, provides an approach that is mathematically and computationally guaranteed to reliably solve difficult nonlinear equation solving and global optimization problems, such as those that arise in chemical process modeling. The most significant drawback of the currently used interval methods is the potentially high computational cost that must be paid to obtain the mathematical and computational guarantees of certainty. New methodologies are described here for improving the efficiency of the interval approach. In particular, a new hybrid preconditioning strategy, in which a simple pivoting preconditioner is used in combination with the standard inverse-midpoint method, is presented, as is a new scheme for selection of the real point used in formulating the interval-Newton equation. These techniques can be implemented with relatively little computational overhead, and lead to a large reduction in the number of subintervals that must be tested during the intervalNewton procedure. Tests on a variety of problems arising in chemical process modeling have shown that the new methodologies lead to substantial reductions in computation time requirements, in many cases by multiple orders of magnitude.

A strategy is described for using linear programming (LP) to bound the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method for deterministic global optimization. An implementation of this technique is described in detail, and several important issues are considered. These include selection of the interval corner required by the LP strategy, and determination of rigorous bounds on the solutions of the LP problems. The impact of using a local minimizer for updating the upper bound on the global minimum in this context is also considered. The procedure based on these techniques, LISS LP, is demonstrated using several global optimization problems, with focus on problems arising in chemical engineering. Problems with a very large number of local optima can be effectively solved, as well as problems with a relatively large number of variables.

Abstract. In many applications of interest in chemical engineering it is necessary to deal with nonlinear models of complex physical phenomena, on scales ranging from the macroscopic to the molecular. Frequently these are problems that require solving a nonlinear equation system and/or finding the global optimum of a nonconvex function. Thus, the reliability with which these computations can be done is often an important issue. Interval analysis provides tools with which these reliability issues can be addressed, allowing such problems to be solved with complete certainty. This paper will focus on three types of applications: 1) Parameter estimation in the modeling of phase equilibrium, including the implications of using locally vs. globally optimal parameters in subsequent computations; 2) Nonlinear dynamics, in particular the location of equilibrium states and bifurcations of equilibria in ecosystem models used to assess the risk associated with the introduction of new chemicals into the environment; 3) Molecular modeling, with focus on transition state analysis of the diffusion of a sorbate molecule in a zeolite. 1.

This project aims at the development of a flexible modeling system for the specification of models for large-scale numerical work in optimization, data analysis, and partial differential equations. Its input should be provided in a form natural for the working mathematician, while the choice of the numerical solvers and the transformation to the format required by the solvers is done by the interface system. The input format should combine the simplicity of LaTeX source code with the semantic conciseness and modularity of current modeling languages such as AMPL, and it should be as close as possible to the mathematical language people use to explain and communicate their models in publications and lectures. In order that the system is useful for the intended applications, interfaces translating the model formulated in the proposed system into the input required for current state of the art solvers, and into the dominant current modeling languages are needed and shall be provided. Moreover, certain shortcomings of the current generation of modeling languages, such as the lack of support for the correct treatment of uncertainties and rounding errors, shall be overcome. The experience gained in this project will be useful in future work in the more general context