The application of dynamic programming to optimal inventory control

This paper concerns a class of deterministic impulse control problems, arising in inventory control. A notable feature of the problem formulation is the presence of an end-point constraint. In consequence, the value function may be discontinuous. Viability theory provides a characterization of the value function as the unique lower semicontinuous solution to a Bensoussan–Lions type quasi-variational inequality (QVI), suitably interpreted for nondifferentiable, extended valued functions. Yet there are few examples in the literature of the use of this analytical machinery. This paper provides such an example. The example, which concerns a problem for which the value function is neither everywhere finite valued nor continuously differentiable on the interior of its effective domain, illustrates what is involved in calculating subdifferentials and checking satisfaction of QVI (in a generalized sense). This paper also provides a summary of the underlying theory, and gathers in the Appendix proofs of key results.