  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.
