  In this paper we study a periodic-review single-commodity setup-cost
inventory model with backorders and holding/backlog costs satisfying
quasiconvexity assumptions. We show that the Markov decision process for this
inventory model satisfies the assumptions that lead to the validity of
optimality equations for discounted and average-cost problems and to the
existence of optimal $(s,S)$ policies. In particular, we prove the
equicontinuity of the family of discounted value functions and the convergence
of optimal discounted lower thresholds to the optimal average-cost one for some
sequences of discount factors converging to $1.$ If an arbitrary nonnegative
amount of inventory can be ordered, we establish stronger convergence
properties: (i) the optimal discounted lower thresholds $s_\alpha$ converge to
optimal average-cost lower threshold $s;$ and (ii) the discounted relative
value functions converge to average-cost relative value function. These
convergence results previously were known only for subsequences of discount
factors even for problems with convex holding/backlog costs. The results of
this paper hold for problems with deterministic positive lead times.

Optimization and Control