Increasing Lipschitz continuous maximizers of some dynamic programs

Abstract

Conditions are presented for the existence of increasing and Lipschitz continuous maximizers in a general one-stage optimization problem. This property results in substantial numerical savings in case of a discrete parameter space. The one-stage result and properties of concave functions lead to simple conditions for the existence of optimal policies, composed of increasing and Lipschitz continuous decision rules, for several dynamic programs with discrete state and action space, in which case discrete concavity plays a dominant role. One of the examples, a general multi-stage allocation problem, is considered in detail. Finally, some known results in the case of a continuous state and action space are generalized.