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

Abstract

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.

20 pages, 6 figures (including tables), 1 algorithm, a convergence proof

References in corpus (1)

• The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis