  Conventional inverse optimization inputs a solution and finds the parameters
of an optimization model that render a given solution optimal. The literature
mostly focuses on inferring the objective function in linear problems when
acceptable solutions are provided as input. In this paper, we propose an
inverse optimization model that inputs several accepted and rejected solutions
and recovers the underlying convex optimization model that can be used to
generate such solutions. The novelty of our model is three-fold: First, while
most literature focuses on inferring the objective function, we focus on
inferring the feasible region. Second, our model can infer the constraints of
general convex optimization models. Third, the proposed model learns from
accepted (good) and rejected (bad) observations in inferring the constraint
set. The resulting inverse model is a mixed-integer nonlinear problem that is
complex to solve. To mitigate the inverse problem complexity, we employ
variational inequalities and the theoretical properties of the solutions to
derive a reduced formulation that retains the complexity of its forward
counterpart. We demonstrate that our inverse model can utilize a subset of past
good and bad treatment plans to infer planning criteria that can lead to nearly
guaranteed clinically acceptable plans for future patients.
