New results on verified global optimization

Abstract

Abstract--ZusammenfassungNew Results on Verified Global Optimization. We study the influence of different strategies for selecting the next box for subdivision in an interval branch and bound method for global optimization. A theoretical result establishes the optimality of a 'best-first' strategy. Moreover, we define a multisection algorithm for subdivision of boxes. Different possibilities for the choice of bisection directions and the number of bisections are compared by numerical results showing that multisection is usually superior to bisection. In addition, we identify crucial properties of a division strategy which yield convergence of an algorithm with multisection. AMS Subject Classification: 65K05.Key words: Global optimization, interval arithmetic, branch and bound principle, best-first strategy, multisection. Neue Ergebnisse bei der verifizierten globalen

Interval algorithm for global numerical optimization

“…Berner suggested that Sel. 1 is the better of the two, since in this criterion the intervals can rarely be subdivided unnecessarily (Berner 1996). In 2001, Casado et al presented a heuristic rejection criterion on the basis of which the interval algorithm implements more efficiently if the boxes with the largest pf * t values are divided at each iteration (Casado et al , 2003.…”

Solving a Huff-like competitive location and design model for profit maximization in the plane

“…The most widely used, and the one we have chosen, is to select the box with the maximal upper bound PðyÞ (known as Moore-Skelboe rule [43]). Berner [1] suggested that it is the most suitable box to choose from the working list, since in this way boxes will rarely be subdivided unnecessarily. However, other criteria [5,6,34] may be better from a computational point of view.…”

Rigorous Global Search: Continuous Problems

“…78-81], [77, §8.8]' and for optimization in [20,199]. Bisection usually proceeds with the following algorithm.…”

“…Berner [20] improves on parallel load balancing schemes of Madsen, Eriksson, and Moore/Hansen/Leclerc. She also uses a variant of "bisection" in which the two largest coordinate directions according to the scaling (5.1) are bisected, thus producing four boxes rather than two.…”