(216f) Rigorous Global Optimization for Dynamic Systems Subject to Path Constraints

Optimization problems involving dynamic systems arise in many areas of engineering and science. In the field of chemical engineering,applications include reaction parameter estimation from time series data and determination of optimal operating profiles for batch and semi-batch processes. In the latter context, industrial practice often requires that reactions and other dynamic processes be operated to satisfy certain safety and/or quality control concerns over the entire operating time. Optimization problems in such systems are thus subject to appropriate path constraints, which state that the operating trajectory may never enter regions of the state space that could cause process safety or product quality problems. Classical local optimization techniques, combined with traditional numerical integration schemes may be insufficient to solve these path-constrained optimization problems rigorously. Classical local optimization techniques provide no guarantee of finding the global optimum when problems exhibit multiple local optima. Moreover, when integrating with traditional numerical schemes, path constraints can only be examined at the end an integration step, not during the entire time interval of the step. To address the issue of multiple local optima, various deterministic global optimization approaches have been developed [1-5]. However, these approaches all focus on the case of unconstrained problem. In contrast, relatively little work has been done on rigorous global optimization for problems with path constraints.

In this study, we present a new strategy for the rigorous global optimization of dynamic systems with inequality path constraints, based on a control parameterization approach. This method is based on Taylor-model constraint propagation [4-6] under a branch-and-reduce framework [1,4,7,8]. A key feature of the method is the use of a validated solver for parametric ODEs (VSPODE), which is used to produce guaranteed bounds on the solutions of dynamic systems with interval-valued decision variables (parameters) [9]. VSPODE consists of two phases applied at each integration step. In the first phase, existence and uniqueness of the solution are proven, and a coarse enclosure of the solution for the entire integration step is computed. In the second phase, a tighter enclosure of the solution is computed and bounded by Taylor models, which are symbolic (algebraic) functions of the decision variables.

Our new strategy provides three main advantages in this application. First, by extracting information from the first phase of the VSPODE algorithm, the path constraints can be tested over the entire time interval of an integration step, thus providing rigorous assurance that the path constraints are satisfied over the entire time horizon [10]. Second, the use of Taylor models to represent bounds provides the power to efficiently obtain rigorous and very tight path enclosures, in comparison to other commonly used bounding methods. Third, because the Taylor model method used provides an explicit algebraic representation of the state trajectories and the objective function in terms of the decision variables, constraint propagation techniques can be exploited to greatly reduce the search space of the decision variables. The computational aspects of this new approach will be demonstrated through application to several example problems.

References

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