To unify and generalize the branch-and-bound method used in operations research and the heuristic search method used in artificial intelligence, a formal description of a branch-and-bound procedure is presented under the assumption that the problem to be solved is given in the form of a discrete decision process (ddp) Y. This is more general than the previous models in that three types of tests, lower-bound test, dominance test, and equivalence test, are all permitted, and the problem Y (ddp) is usually defined on an infinite domain. After proving the validity of this procedure, necessary and sufficient conditions for finite convergence are derived. Then it is shown that the existence of a branch-and-bound procedure for Y is inherently related to the representation of Y by a positively monotone or positively and strictly monotone sequential decision process (pmsdp or psmsdp), which have been studied in conjunction with dynamic programming. This characterizes the class of problems to which a branch-and-bound procedure is applicable.
