  Markov Decision Processes (MDP) is an useful framework to cast optimal
sequential decision making problems. Given any MDP the aim is to find the
optimal action selection mechanism i.e., the optimal policy. Typically, the
optimal policy ($u^*$) is obtained by substituting the optimal value-function
($J^*$) in the Bellman equation. Alternately $u^*$ is also obtained by learning
the optimal state-action value function $Q^*$ known as the $Q$ value-function.
However, it is difficult to compute the exact values of $J^*$ or $Q^*$ for MDPs
with large number of states. Approximate Dynamic Programming (ADP) methods
address this difficulty by computing lower dimensional approximations of
$J^*$/$Q^*$. Most ADP methods employ linear function approximation (LFA), i.e.,
the approximate solution lies in a subspace spanned by a family of pre-selected
basis functions. The approximation is obtain via a linear least squares
projection of higher dimensional quantities and the $L_2$ norm plays an
important role in convergence and error analysis. In this paper, we discuss ADP
methods for MDPs based on LFAs in $(\min,+)$ algebra. Here the approximate
solution is a $(\min,+)$ linear combination of a set of basis functions whose
span constitutes a subsemimodule. Approximation is obtained via a projection
operator onto the subsemimodule which is different from linear least squares
projection used in ADP methods based on conventional LFAs. MDPs are not
$(\min,+)$ linear systems, nevertheless, we show that the monotonicity property
of the projection operator helps us to establish the convergence of our ADP
schemes. We also discuss future directions in ADP methods for MDPs based on the
$(\min,+)$ LFAs.

Systems and Control;Optimization and Control