Inferring Parameters Through Inverse Multiobjective Optimization

Abstract

Given a set of human's decisions that are observed, inverse optimization has been developed and utilized to infer the underlying decision making problem. 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. In this paper, 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, which is important when a precise inference on every single decision maker is practically unnecessary or infeasible.Numerical results on a large number of experiments are reported to confirm the effectiveness of our unique inverse optimization model and the computational efficacy of the developed algorithms.

Wasserstein Distributionally Robust Inverse Multiobjective Optimization

“…For example, outliers in a limited amount of decisions would render the empirical distribution of decisions deviate from the true distribution, and thus significantly weaken the predictive power of the IMOP estimator. We mention that this issue is not unique to IMOP model and one can observe similar findings in other inverse optimization models [2,3,10,11,12,5,13,8,6,4,14].…”

“…Our work is most related to [8,7] that propose a general framework of using inverse multiobjective optimization to infer the objective functions or constraints of the multiobjective DMP, based on observations of Parete optimal solutions. Research in [8,7] presents the framework of empirical risk minimization for this unsupervised learning task, which generally works well when there are few uncertainties in the model, data or hypothetical parameter space.…”

“…Our work is most related to [8,7] that propose a general framework of using inverse multiobjective optimization to infer the objective functions or constraints of the multiobjective DMP, based on observations of Parete optimal solutions. Research in [8,7] presents the framework of empirical risk minimization for this unsupervised learning task, which generally works well when there are few uncertainties in the model, data or hypothetical parameter space. In contrast, we believe that those uncertainties root inherently in the applications of inverse multiobjective optimization, and we aim to hedge against their influences by adopting the distributionally robust optimization paradigm based on Wasserstein metric.…”

“…In the real world, it is basically ubiquitous that decisions are trade-offs among multiple criteria. Hence, inverse optimization considering multiobjective optimization functions, referred to as inverse multiobjective optimization problem (IMOP), provides a more appropriate and compelling tool to learn humans' decision making scheme or emulate their behaviors [7,8]. Specifically, the learner observes a set of Pareto optimal decisions, i.e., {y i } i∈[N ] from the following multiobjective optimization problem (MOP):…”

Generalized Inverse Optimization through Online Learning

“…Indeed, decision makers probably do not have exact information regarding their own decision making process [1]. To bridge that discrepancy, inverse optimization has been proposed and received significant research attention, which is to infer or learn the missing information of the underlying decision models from observed data, assuming that human decision makers are rationally making decisions [2,3,4,5,1,6,7,8,9,10,11]. Nowadays, extending from its initial form that only considers a single observation [2,3,4,5] with clean data, inverse optimization has been further developed and applied to handle more realistic cases that have many observations with noisy data [1,6,7,9,10,11].…”