(118b) Improving the Handling of Interval Bounds during Dynamic Global Optimization

In chemical engineering applications such as safety verification, nonconvex optimization problems must be solved to guaranteed global optimality, to within a specified tolerance. Typical methods for deterministic global optimization compute crucial bounding information by minimizing convex relaxations of objective functions. However, generating useful convex relaxations can be difficult if process models include embedded systems of ordinary differential equations (ODEs), since established reachable-set methods for ODEs are impressive yet still somewhat limited in scope and fidelity. These dynamic relaxation methods are designed to compute convex relaxations for ODE solutions in tandem with interval bounds, by solving an auxiliary relaxation ODE system. By analogy, in the well-known McCormick relaxation approach for constructing convex relaxations of composite functions, these relaxations are also computed in tandem with constant interval bounds, which provide useful range estimates for intermediate variables.

This presentation presents two related refinements [1,2, and under review] to state-of-the-art approaches for computing convex relaxations of ODE solutions, with the ultimate goal of aiding methods for dynamic global optimization. Both of these refinements focus on the relationship between ODE solution relaxations and their related interval bounds. First, we use knowledge of the interval bounds’ correctness to improve the tightness of related ODE solution relaxations, by steering them away from violating the bounds, while maintaining convexity and relaxation validity. Next, we take mathematical techniques that were used to establish correctness of ODE solution relaxations, and adapt these to develop new types of interval bounds on ODE solutions. For example, we obtain new effective interval bounds based on the edge-concave relaxations of Hasan [3]. Numerical examples are presented, and implications are discussed.

References

[1] H Cao and KA Khan, Proceedings of ADCHEM 2021. doi:10.1016/j.ifacol.2021.08.306

[2] H Cao, Ph.D. thesis, McMaster University, 2021. http://hdl.handle.net/11375/27322

[3] MMF Hasan, J. Glob. Optim., 2018. doi:10.1007/s10898-018-0646-x