Assume that a decision-maker’s uncertain behavior is observed. We develop a an inverse optimization framework to impute an objective function that is robust against misspecifications of the behavior. In our model, instead of considering multiple data points, we consider an uncertainty set that encapsulates all possible realizations of the input data. We adopt this idea from robust optimization, which has been widely used for solving optimization problems with uncertain parameters. By bringing robust and inverse optimization together, we propose a robust inverse linear optimization model for uncertain input observations. We aim to find a cost vector for the underlying forward problem such that the associated error is minimized for the worst-case realization of the uncertainty in the observed solutions. That is, such a cost vector is robust in the sense that it protects against the worst misspecification of a decision-maker’s behavior.

As an example, we consider a diet recommendation problem. Suppose we want to learn the diet patterns and preferences of a specific person and make personalized recommendations in the future. The person’s choice, even if restricted by nutritional and budgetary constraints, may be inconsistent and vary over time. Assuming the person’s behavior can be represented by an uncertainty set, it is important to find a cost vector that renders the worst-case behavior within the uncertainty set as close to optimal as possible. Note that the cost vector can have a general meaning and may be interpreted differently depending on the application (e.g., monetary cost, utility function, or preferences). Under such a cost vector, any non-worst-case diet will thus have a smaller deviation from optimality.

We then test our models in the setting of a diet problem on a data-set obtained from NHANES; The data-set is accessible via the link bellow: