  Value-function-based methods have long played an important role in
reinforcement learning. However, finding the best next action given a value
function of arbitrary complexity is nontrivial when the action space is too
large for enumeration. We develop a framework for value-function-based deep
reinforcement learning with a combinatorial action space, in which the action
selection problem is explicitly formulated as a mixed-integer optimization
problem. As a motivating example, we present an application of this framework
to the capacitated vehicle routing problem (CVRP), a combinatorial optimization
problem in which a set of locations must be covered by a single vehicle with
limited capacity. On each instance, we model an action as the construction of a
single route, and consider a deterministic policy which is improved through a
simple policy iteration algorithm. Our approach is competitive with other
reinforcement learning methods and achieves an average gap of 1.7% with
state-of-the-art OR methods on standard library instances of medium size.
