Inverse optimization is a powerful paradigm for learning preferences and restrictions that explain the behavior of a decision maker, based on a set of external signal and the corresponding decision pairs. However, most inverse optimization algorithms are designed specifically in a batch setting, where all data is available in advance. As a consequence, there has been rare use of these methods in an online setting that is more suitable for real-time applications. To change such a situation, we propose a general framework for inverse optimization through online learning. Specifically, we develop an online learning algorithm that uses an implicit update rule which can handle noisy data.

We also note that the majority of existing studies assumes that the decision making problem is with a single objective function, and attributes data divergence to noises, errors or bounded rationality, which, however, could lead to a corrupted inference when decisions are tradeoffs among multiple criteria. We take a data-driven approach and design a more sophisticated inverse optimization formulation to explicitly infer parameters of a multiobjective decision making problem from noisy observations. This framework, together with our mathematical analyses and advanced algorithm developments, demonstrates a strong capacity in estimating critical parameters, decoupling interpretable components from noises or errors, deriving the denoised optimal decisions, and ensuring statistical significance. In particular, for the whole decision maker population, if suitable conditions hold, we will be able to understand the overall diversity and the distribution of their preferences over multiple criteria.

Additionally, we propose a distributionally robust approach to inverse multiobjective optimization. Specifically, we study the problem of learning the objective functions or constraints of a multiobjective decision making model, based on a set of observed decisions. In particular, these decisions might not be exact and possibly carry measurement noises or are generated with the bounded rationality of decision makers. We use the Wasserstein metric to construct the uncertainty set centered at the empirical distribution of these decisions. We show that this framework has statistical performance guarantees. We also develop an algorithm to solve the resulting minmax problem and prove its finite convergence.