Recommendations as Designed Experiments: Active Inverse Optimization for Identifiable Preference Learning under Hard Constraints

Inverse optimization (IO) personalizes constrained recommendations, for instance, diets under clinical guidelines, by recovering the objective that renders observed decisions optimal, then re-optimizing. It has been shown that passive observation does *not* necessarily identify the objective, but instead recovers a polyhedral cone of parameters and more data from the same behavior cannot shrink it. However, a recommender system can be viewed as a designed experiment that needs to offer targeted interventions that are informative while maintaining or improving user's baseline goals (e.g., dietary goals). We formalize this as active, intervention-aware, online IO where accept/reject responses cut a tracked admissible set of preferences; slates composed within an $\varepsilon$-indifference family of guaranteed-improving options constructively satisfy the excitation condition that passive data violate; and a randomization-identified exposure term corrects the bias the system injects into its own data. We validate this system using a closed-loop simulation that shows 100% non-worsening recommendations while reducing intervention-biased preference error from 0.20 to 0.13 by round 60.