🤖 AI Summary
This work addresses the challenge of accurately identifying optimal policies and active constraints when both are unknown. It proposes a unified simulation-grid-based dual framework that integrates Fenchel duality, Doob martingale compensation, complementary slackness, and occupancy-measure weighting. By decomposing residuals via conditional budget identities and leveraging Bellman curvature, the method constructs a tight policy region without requiring a reference solution. It simultaneously estimates policy error, certifies active constraint facets, and provides joint verification of value bounds and policy distance. Empirical results demonstrate its ability to achieve full coverage in auditing external policy errors, deliver zero false positives in active facet detection, maintain tightness in 50-dimensional asset stress tests, and reveal that dual-learning accuracy becomes the performance bottleneck in high dimensions.
📝 Abstract
Neural and numerical policy solvers can produce feasible controls even when the optimal rule and its binding constraints are unavailable. A primal-dual bracket certifies value loss, but it does not locate the optimal policy or explain which constraints genuinely bind. We show that, on the same declared simulation grid, one bracket can support both conclusions. For polyhedral controls, an exact conditional budget identity rewrites the residual as a pathwise nonnegative terminal Fenchel defect plus date-by-constraint complementary-slackness terms. A canonical Doob compensation removes the budget martingale that obscures small residuals. Bellman-primitive curvature conditions then yield an occupancy-weighted policy region with the sharp O(sqrt(G)) radius, while a paired constraint relaxation lower-bounds the optimal multiplier and certifies a binding face. A finite-sample resolution theorem quantifies the path budget needed to certify a target policy tolerance or face. Locked one-asset and two-asset audits cover every external policy error and make no false face declaration. An exact-wrapper stress test remains tight through 50 assets, while a separate state-dependent pilot identifies learned-dual tightness as the high-dimensional bottleneck. Reference solutions enter only after certification.