Bayesian optimization for congestion pricing problems: A general framework and its instability

In this study, we proposed a generic Bayesian optimization (BO) framework to solve congestion pricing problems. In the BO framework, the Gaussian process (GP) serves as a surrogate model to approximate the highly nonlinear and expensive-to-evaluate objective functions. This study reveals that GP exhibits an instability phenomenon, which inherently limits the accuracy of BO. We investigate the sources and influences of instability from the perspective of error analysis, and then propose an improved GP (IGP) model to address the instability issue. The associated improvements are twofold: matrix inversion and matrix multiplication. A tailored preconditioner is developed to reduce the matrix inversion errors. To address multiplication errors, a tailored dot product algorithm in conjunction with a GP reformulation scheme is proposed. To validate the proposed models and methods, a link-based second-best congestion pricing problem is considered as an example. The results indicate that, in comparison to benchmark approaches (the sensitivity analysis method and genetic algorithm), the proposed BO framework shows higher computational efficiency and solution accuracy. With modifications on GP, the instability phenomenon is substantially mitigated in several instances, hence enhancing the accuracy of the BO framework.