Theoretical Justi cation of a Heuristic Subbox Selection Criterion

The most widely used guaranteed methods for global optimization are probably the interval-based branch-and-bound techniques. In these techniques, we start with a single box { the entire function domain { as a possible location of the global minimizer points, and then, in each step, subdivide some of the boxes, use interval computations to compute the enclosure F (X); F(X)] f(X) of the range f(X) of the objective function f(x) on each new sub-box X, and, based on these computations, eliminate the boxes which cannot contain a global minimizer point. The computational eeciency of these methods strongly depends on which boxes we select for subdivision. Traditionally, for subdivision , the algorithms selected a box with the smallest value of F(X). Recently, it was shown that the algorithm converges much faster if we select, instead, a box with the largest possible value of the ratio e f ? F(X) F(X) ? F(X) ; where e f is a current upper bound on the actual global minimum. In this paper, we give a theoretical justiication for this empirical criterion. Namely, we show that: { rst, this criterion is the only one that is invariant w.r.t. some reasonable symmetries; and { second, that this criterion is optimal in some reasonable sense.