Ordered Solutions for Dynamic Programs

Suppose X indexes a collection of optimization problems in Rn and δ*x is an optimal program for the problem indexed by x ∈ X. Sufficient conditions are presented for two other families of optimization problems to possess optimal programs δ0x and δ0x for the respective problems when x ∈ X such that δ0x ≤ δ*x ≤ δ0x for all x ∈ X. This result is then applied to obtain bounds on an optimal policy in the class of dynamic programs which are bounded, monotone contractions.