Maximum Optimality Margin: A Unified Approach for Contextual Linear Programming and Inverse Linear Programming

The predict-then-optimize problem considers a learning problem under a decision making context where the output of a machine learning model serves as the input of a downstream optimization problem (e.g. a linear program). The ultimate goal of the learner is to prescribe a decision/solution for the downstream optimization problem using directly the input (variables) of the machine learning model but without full observation of the input of the optimization problem. A similar problem formulation was also studied as prescriptive analytics (Bertsimas and Kallus, 2020) and contextual linear programming (Hu et al., 2022). While Elmachtoub and Grigas (2022) justifies the importance of leveraging the optimization problem structure when building the machine learning model, the existing efforts on exploiting the optimization structure have been largely inadequate. In this paper, we delve deeper into the structural properties of the optimization problem and propose a new approach called maximum optimality margin which builds a max-margin learning model based on the optimality condition of the downstream optimization problem. More importantly, our approach only needs the observations of the optimal solution in the training data rather than the objective function, thus it draws an interesting connection to the inverse optimization problem. The connection gives a new shared perspective on both the predict-then-optimize problem and the inverse optimization problem, and our analysis reveals a scale inconsistency issue that arises practically and theoretically for many existing methods.