Enhancing the Efficiency of Solving Global Dynamic Optimization Problems by Improving the Calculation of State Relaxations
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Ye, Jason
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Abstract
The ability to solve global dynamic optimization (GDO) problems computationally is crucial in modeling and optimizing processes that change over time. For instance, in chemical manufacturing, one might be interested in determining how much raw materials to inject into a batch reactor to maximize the product that can sell for the most money, subject to constraints in the form of such physical laws as transient mass and energy balances. In air navigation, it may be of interest to find the thrust of an aircraft to minimize its traveling distance as it moves over time to its destination, subject to the constraint that it does not collide with an obstacle. Spatial branch-and-bound (B&B) is a technique used to solve GDO problems. Currently, B&B can only solve, within realistic time, GDO problems with a limited number of input variables, namely up to around 10 state and 3 decision variables in roughly 3 hours. However, this falls far short of the capability needed to solve GDO problems computationally at the real scale, which contains far more such variables. Yet, even if we can solve GDO problems containing tens of state and decision variables in a matter of hours, we will have already been able to solve many more GDO problems using branch-and-bound than we are currently able to, including problems dealing with reaction kinetics. Thus, there is a critical need to improve B&B's efficiency in solving GDO problems. A useful technique for enhancing the efficiency of B&B is improving the accuracy or bounding tightness of a convex program whose optimal objective value underestimates that of the original optimization problem. The calculation of such convex programs, known as convex relaxations, is a major step in B&B. By tightening these relaxations around their real functions, their bounding accuracy improves, which leads to a shorter computational time taken by B&B in solving the GDO problem at hand. To that end, this thesis presents multiple methods for improving the calculation of relaxations in a GDO problem, in which enhancing their bounding tightness is a key goal. By improving the efficiency of solving GDO problems, we can come closer to the goal of solving these problems at the real scale of interest, thus serving as an alternative to hands-on experimentation for answering the kinds of questions targeted by GDO problem solving.
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Date
2024-04-28
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Dissertation