Optimization Methods for Mixed-Integer Control Problems in Complex Systems

Abstract

Applying sophisticated optimization methods to control problems in various systems offers a structured approach for achieving system efficiency and reliability, while ensuring the feasibility of physics-based constraints. This dissertation primarily focuses on developing optimization algorithms for control problems in complex systems with mixed-integer variables and parameter uncertainties. In Chapter 1, we provide a detailed introduction about the problem motivation and summarize our problems and contributions. Specifically, we develop new optimization models and algorithms for network-based traffic control and binary control in quantum systems, detailed as follows. Traffic signal control is an effective way of mitigating traffic congestion, where its complexity escalates in large cities due to numerous intersections and varying traffic conditions. Advancements in communication technologies within transportation infrastructures have made distributed traffic signal control at the network level possible and worth investigating. In Chapter 2, we build a stochastic optimization model for network-level traffic signal control under traffic demand uncertainty and solve it using decentralized algorithms. We compare the results with state-of-the-art traffic control methods via testing instances of real-world traffic networks and data. Quantum computing and quantum systems provide a novel way of significantly accelerating computation and their operations largely rely on quantum pulse control optimization to attain desired states or status. The optimization of binary quantum pulse control can potentially enhance the performance of classical quantum variational techniques and improve solution quality. In Chapter 3, we study a discrete-valued binary quantum control problem and propose an algorithmic framework to solve the problem with penalty on switches. In Chapter 4, we explore the binary quantum control problem in a continuous time horizon by optimizing both control functions and control time intervals. Moreover, the time-varying noise in quantum systems and the wide use of inhomogeneous quantum ensembles highlight the need for quantum controls considering uncertainties. In Chapter 5, we develop a stochastic optimization model for the binary quantum control problem with uncertain Hamiltonian controllers and solve it using gradient-based methods with rounding algorithms. We test the performance of our approaches and benchmark with state-of-the-art control methods in all three chapters. Lastly, in Chapter 6, we conclude the dissertation and state future research directions.