Approximate Dynamic Programming based on Projection onto the (min,+) subsemimodule

We develop a new Approximate Dynamic Programming (ADP) method for infinite horizon discounted reward Markov Decision Processes (MDP) based on projection onto a subsemimodule. We approximate the value function in terms of a $(\min,+)$ linear combination of a set of basis functions whose $(\min,+)$ linear span constitutes a subsemimodule. The projection operator is closely related to the Fenchel transform. Our approximate solution obeys the $(\min,+)$ Projected Bellman Equation (MPPBE) which is different from the conventional Projected Bellman Equation (PBE). We show that the approximation error is bounded in its $L_\infty$-norm. We develop a Min-Plus Approximate Dynamic Programming (MPADP) algorithm to compute the solution to the MPPBE. We also present the proof of convergence of the MPADP algorithm and apply it to two problems, a grid-world problem in the discrete domain and mountain car in the continuous domain.