Review of branch-and-bound framework

In this chapter, we present a general and effective approach to solve NP-hard or NP-complete problems, despite their potential exponential worst-case complexity. Although the complexity of this approach is nonpolynomial, when designed appropriately, this approach may still be a viable approach to solve many problems that arise in practice, particularly when their problem sizes are not terribly large.

The approach that we describe is called branch-and-bound [113]. It is a powerful general-purpose framework to solve nonconvex programming problems. Such a framework aims to obtain a (1 − ∊)-optimal solution for a small given ∊ ≥ 0. Obviously, the smaller ∊ is, the higher is the complexity. Thus, we need to select a suitable ∊ based on the optimality requirement. The efficiency of the algorithm depends on its ability to correctly remove large portions of the solution space during each iterative step within the process of identifying the best solution. As a result, we progressively focus on smaller and smaller portions of the solution space and thus we are able to find a (1 − ∊)-optimal solution much faster than a brute-force exhaustive search.

We now describe the branch-and-bound framework for a minimization problem (the procedure for a maximization problem is similar). Initially, we need to determine the solution space for our problem, including the set of feasible values for each of the optimization variables. The framework consists of two steps, namely, the bounding step and the branching step.