Extending the reduction from polytopic RMDPs to RMDPs with non-polytopic uncertainty sets may face several limitations and challenges. One major challenge is the complexity of representing and manipulating non-polytopic uncertainty sets. Polytopic uncertainty sets have a simple geometric structure that allows for efficient computation and optimization. Non-polytopic uncertainty sets, on the other hand, can be more complex and may not have a straightforward representation in terms of vertices or convex hulls. This complexity can make it challenging to apply the same reduction techniques used for polytopic RMDPs.
Another limitation is the computational complexity of dealing with non-polytopic uncertainty sets. Algorithms designed for polytopic uncertainty sets may not be directly applicable to non-polytopic sets, as the computational complexity of solving RMDPs can increase significantly with the complexity of the uncertainty sets. This could lead to challenges in developing efficient algorithms and may require novel approaches to handle the increased complexity.
Furthermore, the theoretical foundations and results established for polytopic RMDPs may not directly translate to RMDPs with non-polytopic uncertainty sets. The properties and characteristics of non-polytopic uncertainty sets may introduce new challenges and considerations that need to be addressed in the algorithm design and analysis.