The Principle of Optimality in Dynamic Programming with Returns in Partially Ordered Sets

A general sequential model is defined where returns are in a partially ordered set. A distinction is made between maximal (nondominated) returns and greatest returns. We show that Bellman's principle of optimality is valid with respect to maximal returns and it leads to an algorithm to approximate these returns. We argue that significantly greater effort is needed to apply this algorithm to maximal returns than to greatest returns.