Integrating Heuristics and Approximations into a Branch and Bound Framework

Algorithms for solving large-scale optimization problems often use heuristics and approximations to achieve a solution quickly, however there is often little or no information as to the quality of the solution. We integrate heuristics and approximations into a branch and bound framework to take advantage of obtaining a solution quickly, while using the framework to prune regions that do not contain an optimal solution, and provide an optimality gap. Three examples are cast into this framework. First, we describe a Rollout Algorithm with Branch-and-Bound (RA-BnB) that embeds an approximate dynamic program into a branch and bound framework to address a challenging resource allocation problem in population disease management. Second, we describe a Vehicle Routing and Scheduling Algorithm (VeRSA) that embeds an easily calculated index, as is commonly used in scheduling, to dynamically search and prune a branch and bound tree. Third, we describe a Probabilistic Branch and Bound algorithm (PBnB) that uses a statistical sampling method to obtain confidence interval bounds that are embedded into a tree to probabilistically prune regions of the tree. These three, apparently different, methods share commonalities that make use of heuristics and approximations to generate a “near-optimal” solution quickly, and also provide information on the quality of the solution by providing an optimality gap. Lessons learned on implementation decisions and how to balance computation in the context of these three problems are discussed.