Inverse Optimization via Learning Feasible Regions

We study inverse optimization (IO), where the goal is to use a parametric optimization program as the hypothesis class to infer relationships between input-decision pairs. Most of the literature focuses on learning only the objective function, as learning the constraint function (i.e., feasible regions) leads to nonconvex training programs. Motivated by this, we focus on learning feasible regions for known linear objectives, and introduce two training losses along with a hypothesis class to parameterize the constraint function. Our hypothesis class surpasses the previous objective-only method by naturally capturing discontinuous behaviors in input-decision pairs. We introduce a customized block coordinate descent algorithm with a smoothing technique to solve the training problems, while for further restricted hypothesis classes, we reformulate the training optimization as a tractable convex program or mixed integer linear program. Synthetic experiments and two power system applications including comparisons with state-of-the-art approaches showcase and validate the proposed approach.

Many real-world systems make complex decisions within strict boundaries, e.g., power grid operators who must balance electricity supply while respecting physical limitations. When experts observe such decisions, they often want to understand what invisible rules or constraints are influencing them. We develop a method to reverse-engineer these hidden constraints from observed decisions, assuming we already know the decision-makers’ goals. Unlike earlier approaches that are dedicated to learn the objectives, our method specifically focuses on learning the constraints that shape feasible decisions. A key innovation of our work is its ability to handle situations where small changes in conditions lead to large shifts in behavior, i.e., the ability to learn a potentially discontinuous behavior. We demonstrate our method's effectiveness through both synthetic experiments and real power system applications, where it outperforms existing approaches and offers valuable insights into complex decision-making processes.