Dynamic optimization determines values for input/control profiles, real valued parameters, initial conditions and/or boundary conditions of a dynamic system that optimize its performance over some period of time according to a specified metric. Dynamic optimization problems appear in almost every aspect of modern chemical engineering. The dynamic optimization problems encountered in chemical engineering often exhibit multiple suboptimal local minima. Conventional approaches for the solution of dynamic optimization problems can only guarantee locating local minima, which may be suboptimal. Suboptimality can have direct economic, safety and environmental impacts if a suboptimal solution is implemented on a real system. This project will develop theory, numerical methods and software that can guarantee locating a global solution of a dynamic optimization problem within a finite number of iterations.
Intellectual Merit Previous research by the PI supported by the NSF has created global optimization theory and algorithms for optimization problems embedding linear time varying ordinary differential equations (ODEs) and nonlinear ODEs. The approach has been demonstrated for relatively small dynamic systems and up to about ten degrees of freedom in the optimization problem. The purpose of this research is to develop the theory and algorithms further so that optimization problems involving large-scale dynamic systems (possibly 1,000s-10,000s of state variables) and tens of degrees of freedom can be solved to guaranteed global optimality. This would bring a majority of the dynamic optimization problems in chemical engineering within the scope of global optimization methods. A key theoretical and practical issue in this extension is the computation of tight estimates of the image of a parameter set under the solution of nonquasi-monotone differential equations (most chemical engineering applications are nonquasi-monotone). Theory and algorithms based on extensions of classical results in differential inequalities will be used to address this issue. Moreover, many dynamic optimization problems in chemical engineering also have differential-algebraic equations (DAEs) and/or partial differential equations (PDEs) embedded. Theory and algorithms extending the global optimization approach to DAE and PDE embedded systems are planned.
An application of global dynamic optimization is formal safety verification; deterministic global optimization provides a constructive proof that a dynamic system is safe, or guarantees location of a counterexample. However, formal verification is always with respect to a model and does not take into account the fact that there is always a discrepancy between the predictions of a model and the behavior of the corresponding physical system. This is commonly referred to as model uncertainty. Previous research has not considered the issue of model uncertainty in formal safety verification, but this is a potentially critical issue in guaranteeing the safety of a physical system. An approach based on a semi-infinite program with differential equations embedded is proposed to address model uncertainty in safety verification.
Broader Impacts: The growing capability to solve dynamic optimization problems to guaranteed global optimality could have broad practical implications. For example, in the area of process operations there is hope for solving problems such as formal safety verification under uncertainty, the synthesis of integrated batch processes, and the design of major process transients such as start-up and shut-down procedures, using detailed dynamic models. The results of this work will be broadly disseminated through journal articles, publicly distributed software, course curricula and a textbook on global optimization currently being prepared by the PI. Moreover, the software developed through this project will be freely distributed via the Web to academic researchers. The project should provide students many opportunities for multidisciplinary education and research. The large number of female undergraduate and graduate students in the department should help in attracting some of them t o work on this project.
