Working in Reverse: Advancing Inverse Optimization in the Fields of Equilibrium and Infrastructure Modeling

Abstract

Transportation and infrastructure modeling allows us to pursue societal aims such as improved disaster management, traffic flow, and water allocation. Equilibrium programming enables us to represent the entities involved in these applications such that we can learn more about their dynamics. These entities include transportation users and market players. However, determining the parameters in these models can be a difficult task because the entities involved in these equilibrium processes may not be able to articulate or to disclose the parameterizations that motivate them. The field of inverse optimization (IO) offers a potential solution to this problem by taking observed equilibria to these systems and using them to parameterize equilibrium models. In this dissertation, we explore the use of inverse optimization to parameterize multiple new or understudied subclasses of equilibrium problems as well as expand inverse optimization's application to new infrastructure domains. In the first project of our dissertation, our contribution to the literature is to propose that IO can be used to parameterize cost functions in multi-stage stochastic programs for disaster management and can be used in disaster support systems. We demonstrate in most of our experiments that using IO to obtain the hidden cost parameters for travel on a road network changes the protection decisions made on that road network when compared to utilizing the mean of the parameter range for the hidden parameters (also referred to as ``uniform cost''). The protection decisions made under the IO cost parameterizations versus the true cost parameterizations are similar for most of the experiments, thus lending credibility to the IO parameterizations. In the second project of our dissertation, we extend a well-known framework in the IO community to the case of jointly convex generalized Nash equilibrium problems (GNEPs). We demonstrate the utility of this framework in a multi-player transportation game in which we vary the number of players, the capacity level, and the network topology in the experiments as well as run experiments assuming the same costs among players and different costs among players. Our promising results provide evidence that our work could be used to regulate traffic flow toward aims such as reduction of emissions. In the final project of our dissertation, we explore the general parameterization of the constant vector in linear complementarity problems (LCPs), which are mathematical expressions that can represent optimization, game theory, and market models (Gabriel et al., 2012). Unlike the limited previous work on inverse optimization for LCPs, we characterize theoretical considerations regarding the inverse optimization problem for LCPs, prove that a previously proposed IO solution model can be dramatically simplified, and handle the case of multiple solution data points for the IO LCP problem. Additionally, we use our knowledge regarding LCPs and IO in a water market allocation case study, which is an application not previously explored in the IO literature, and we find that charging an additional tax on the upstream players enables the market to reach a system optimal. In sum, this dissertation contributes to the inverse optimization literature by expanding its reach in the equilibrium problem domain and by reaching new infrastructure applications.