Theoretical Justification of a Heuristic Subbox Selection Criterion

Authors

Vladik Kreinovich, The University of Texas at El PasoFollow
Tibor Csendes

Publication Date

8-2001

Comments

UTEP-CS-01-03a.

Published in Central European Journal of Operations Research CEJOR, 2001, Vol. 9, No. 3, pp. 255-265.

Abstract

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 minimum, and then, of each step, subdivide some of the boxes, use interval computations to compute the enclosure [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 the global minimum. The computational efficiency of these methods strongly depends on which boxes we select for sub-division. Traditionally, for sub-division, 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 (f-F-(X))/ (F+(X)-F-(X)), where f is a current upper bound on the actual global minimum. In this paper, we give a theoretical justification for this empirical criterion. Namely, we show that:

first, 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.