Bounded-Error Policy Optimization for Mixed Discrete-Continuous MDPs via Constraint Generation in Nonlinear Programming

We propose the Constraint-Generation Policy Optimization (CGPO) framework to optimize policy parameters within compact and interpretable policy classes for mixed discrete-continuous Markov Decision Processes (DC-MDP). CGPO can not only provide bounded policy error guarantees over an infinite range of initial states for many DC-MDPs with expressive nonlinear dynamics, but it can also provably derive optimal policies in cases where it terminates with zero error. Furthermore, CGPO can generate worst-case state trajectories to diagnose policy deficiencies and provide counterfactual explanations of optimal actions. To achieve such results, CGPO proposes a bilevel mixed-integer nonlinear optimization framework for optimizing policies in defined expressivity classes (e.g. piecewise linear) and reduces it to an optimal constraint generation methodology that adversarially generates worst-case state trajectories. Furthermore, leveraging modern nonlinear optimizers, CGPO can obtain solutions with bounded optimality gap guarantees. We handle stochastic transitions through chance constraints, providing high-probability performance guarantees. We also present a roadmap for understanding the computational complexities of different expressivity classes of policy, reward, and transition dynamics. We experimentally demonstrate the applicability of CGPO across various domains, including inventory control, management of a water reservoir system, and physics control. In summary, CGPO provides structured, compact and explainable policies with bounded performance guarantees, enabling worst-case scenario generation and counterfactual policy diagnostics.