Optimization-Based Robust Permissive Synthesis for Interval MDPs

We present an optimization-based framework for robust permissive synthesis for Interval Markov Decision Processes (IMDPs), motivated by robotic decision-making under transition uncertainty. In many robotic systems, model inaccuracies and sensing noise lead to interval-valued transition probabilities. While robust IMDP synthesis typically yields a single policy and permissive synthesis assumes exact models, we show that robust permissive synthesis under interval uncertainty can be cast as a global mixed-integer linear program (MILP) that directly encodes robust Bellman constraints. The formulation maximizes a quantitative permissiveness metric (the number of enabled state-action pairs), while guaranteeing that every compliant strategy satisfies probabilistic reachability or expected reward specifications under all admissible transition realizations. To address the exponential complexity of vertex-based uncertainty representations, we derive a dualization-based encoding that eliminates explicit vertex enumeration and scales linearly with the number of successors. Experimental evaluation on four representative robotic benchmark domains demonstrates scalability to IMDPs with hundreds of thousands of states. The proposed framework provides a practical and general foundation for uncertainty-aware, flexibility-preserving controller synthesis in robotic systems.