  Consider a problem where a set of feasible observations are provided by an
expert and a cost function is defined that characterizes which of the
observations dominate the others and are hence, preferred. Our goal is to find
a set of linear constraints that would render all the given observations
feasible while making the preferred ones optimal for the cost (objective)
function. By doing so, we infer the implicit feasible region of the linear
programming problem. Providing such feasible regions (i) builds a baseline for
categorizing future observations as feasible or infeasible, and (ii) allows for
using sensitivity analysis to discern changes in optimal solutions if the
objective function changes in the future.
  In this paper, we propose an inverse optimization framework to recover the
constraints of a forward optimization problem using multiple past observations
as input. We focus on linear models in which the objective function is known
but the constraint matrix is partially or fully unknown. We propose a general
inverse optimization methodology that recovers the complete constraint matrix
and then introduce a tractable equivalent reformulation. Furthermore, we
provide and discuss several generalized loss functions to inform the desirable
properties of the feasible region based on user preference and historical data.
Our numerical examples verify the validity of our approach, emphasize the
differences among the proposed measures, and provide intuition for large-scale
implementations. We further demonstrate our approach using a diet
recommendation problem to show how the proposed models can help impute
personalized constraints for each dieter.
