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# Inverse Optimization and Inference models

### UTILITY INFERENCE UNDER UNCERTAINTY

Given an optimal or near-optimal solution, an inverse optimization problem determines objective function parameters of the forward optimization problem such that the solution becomes optimal for the forward problem. However, there is often uncertainty in the observed solutions and hence there is a need to develop robust inverse optimization models.

### A COMBINED INVERSE OPTIMIZATION FOR PERSONALIZED DIET

Many patients who suffer from hypertension or diabetes have to change their diet and lifestyle to control their symptoms and the progress of their illness. However, the diets recommended often follow a one-size-fits-all philosophy and are rarely adjusted to the patients’ taste and preferences. Hence, many patients find it hard to adhere to such diets. We use inverse optimization to find a personalized diet based on a patient’s previous choices.

### DATA-DRIVEN INVERSE OPTIMIZATION TO PERSONALIZED CANCER CARE

The medical decisions of experts, for instance the radiation treatment plan approved by oncologists, can be observed. However, the true underlying criteria based on which they approved or rejected a plan is not known. We develop data-driven inverse optimization models to infer the constraints parameters of an optimization problem to understand such expert-driven decisions better.

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.