Convex Underestimators for Dynamic Optimization Problems

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ABSTRACTPI: Paul I. Barton Institution: MITProposal Number: 0120441The objective of this research is to develop deterministic global optimization algorithms for nonconvex dynamic embedded optimization problems, mixed-integer dynamic optimization problems and nonconvex variational problems. The approaches need to be practically implementable and they should provide theoretical guarantees of locating the global optimum.Research: In the first research task, the PI will develop a convexity theory and convex underestimators for optimization problems with a very general integral objective function and linear time varying dynamic system embedded. With these theoretical foundation, it will be possible to adapt existing deterministic global optimization algorithms for nonconvex nonlinear programs and mixed-integer nonlinear program, hence also addressing mixed-integer dynamic optimization problems. Next, the PI will explore how this composite function approach can be extended to problems with nonlinear dynamic systems embedded. This extension will draw heavily on the theory developed for the linear case. In the final research task, he will explore the construction of convex underestimators for variational and optimal control problems, i.e., convex underestimators on linear spaces of functions. A number of major research challenges exist in the application of these variational convex underestimators, including the numerical generation of rigorous lower bounds from the convex underestimating problems, and how to partition a linear space of function in, for example, a branch-and-bound procedure.Impact: The research will make fundamental contributions via developing a convexity theory for dynamic embedded optimization problems, and developing methods for the construction of convex underestimators for dynamic embedded optimization problems and variational problems. This theory will lead to a series of practical deterministic global optimization algorithms for the solution of these problems. The capability to solve such problems to guaranteed global optimality will have broad practical implications. For example, in the area of process operations there is hope for solving problems such as formal safety verification, 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. Most serious industrial accidents occur during such transient events. Furthermore, the method will have applicability in other engineering disciplines as well as applied mathematics.