FIGURE 1.

Illustration of αBB algorithm in one dimension. The domain is first bisected, and the minima of the convex underestimators in the two subdomains are located at 1 and 2. 1 and 2 are each projected upward onto the function and used as starting points for local minimizations of this function, finding the minima at 3 and 4. 3 is the lower of these, so it is taken as the system upper bound. Note that the right-hand side of the domain has a lower bound at 2, which is higher than the system upper bound at 3; this region is therefore fathomed. Further bisection of the left subdomain occurs. At this point, the upper and lower bounds converge at point 7—this point is higher in energy than 5, but lower in energy than 6, meaning that it represents both the upper and lower bound on the system. This indicates that 7 is the global optimum solution.