Expert-Guided Inverse Optimization for Convex Constraint Inference

Abstract: 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 accepted 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 two-fold: First, we focus on inferring the parameters of the underlying convex feasible region. Second, the proposed model learns the convex constraint set from a set of past observations that are either accepted or rejected by an expert. 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. Using realistic breast cancer patient data, we demonstrate that our inverse model can utilize a subset of past accepted and rejected treatment plans to infer clinical criteria that can lead to nearly guaranteed acceptable treatment plans for future patients.