Convexity and the maximum principle for discrete systems

Halkin has given a derivation of the discrete maximum principle using a convexity requirement. An example given in this paper shows that incorrect results may be obtained when Halkin's convexity requirement is not met. There are, however, systems that do not satisfy the convexity requirement, but for which there is still a maximum principle. The discrete maximum principle is rederived with a requirement, directional convexity, that is weaker than convexity and which considerably extends its applicability. Though convexity has appeared to be basic in the development of optimal control theory, it is only the weaker property of directional convexity which is required for much of the development.