Bisecton by Global Optimization Revisited

In partitioning methods such as branch and bound for solving global optimization problems, the so-called bisection of simplices and hyperrectangles is used since almost 40 years. Bisections are also of interest in finite-element methods. However, as far as we know, no proof has been given of the optimality of bisections with respect to other partitioning strategies. In this paper, after generalizing the current definition of partition slightly, we show that bisection is not optimal. Hybrid approaches combining different subdivision strategies with inner and outer approximation techniques can be more efficient. Even partitioning a polytope into simplices has important applications, for example in computational convexity, when one wants to find the inequality representation of a polytope with known vertices. Furthermore, a natural approach to the computation of the volume of a polytope P is to generate a simplex partition of P, since the volume of a simplex is given by a simple formula. We propose several variants of the partitioning rules and present complexity considerations. Finally, we discuss an approach for the volume computation of so-called H-polytopes, i.e., polytopes given by a system of affine inequalities. Upper bounds for the number of iterations are presented and advantages as well as drawbacks are discussed.