The theoretical and empirical rate of convergence for geometric branch-and-bound methods

  • Authors:

  • Anita Schöbel;Daniel Scholz

  • Affiliations:

  • Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Göttingen, Germany;Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Göttingen, Germany

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2010

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Abstract

Geometric branch-and-bound solution methods, in particular the big square small square technique and its many generalizations, are popular solution approaches for non-convex global optimization problems. Most of these approaches differ in the lower bounds they use which have been compared empirically in a few studies. The aim of this paper is to introduce a general convergence theory which allows theoretical results about the different bounds used. To this end we introduce the concept of a bounding operation and propose a new definition of the rate of convergence for geometric branch-and-bound methods. We discuss the rate of convergence for some well-known bounding operations as well as for a new general bounding operation with an arbitrary rate of convergence. This comparison is done from a theoretical point of view. The results we present are justified by some numerical experiments using the Weber problem on the plane with some negative weights.