Convex Optimization and Online Learning: Their Applications in Discrete Choice Modeling and Pricing

Abstract

The discrete choice model has been an important tool to model customers' demand when facing a set of substitutable choices. The random utility model, which is the most commonly used discrete choice framework, assumes that the utility of each alternative is random and follows a prescribed distribution. Due to the popularity of the random utility model, the probabilistic approach has been the major method to construct and analyze choice models. In recent years, several choice frameworks that are based on convex optimization are studied. Among them, the most widely used frameworks are the representative agent model and the semi-parametric choice model. In this dissertation, we first study a special class of the semi-parametric choice model - the cross moment model (CMM) - and reformulate it as a representative agent model. We also propose an efficient algorithm to calculate the choice probabilities in the CMM model. Then, motivated by the reformulation of the CMM model, we propose a new choice framework - the welfare-based choice model - and establish the equivalence between this framework and the other two choice frameworks: the representative agent model and the semi-parametric choice model. Lastly, motivated by the multi-product pricing problem, which is an important application of discrete choice models, we develop an online learning framework where the learning problem shares some similarities with the multi-product pricing problem. We propose efficient online learning algorithms and establish convergence rate results for these algorithms. The main techniques underlying our studies are continuous optimization and convex analysis.