Abstract
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.
Similar content being viewed by others
References
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)
Horst, R., Pardalos, P.H., Thoai, N.V.: Introduction to Global Optimization, 2nd edn. Kluwer Academic, Dordrecht (2000)
Horst, R.: An algorithm for nonconvex programming problems. Math. Program. 10, 312–321 (1976)
Horst, R.: A note on the convergence of an algorithm for nonconvex programming problems. Math. Program. 19, 237–238 (1980)
Horst, R.: An generalized bisection of n-simplicies. Math. Comput. 66, 691–698 (1997)
Nast, M.: Subdivision of simplices relative to a cutting plane and finite concave minimization. J. Glob. Optim. 9, 1 (1996)
Edelsbrunner, H.: Geometric Algorithms. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A, pp. 699–735. North-Holland, Amsterdam (1993)
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Berlin (1987)
Grünbaum, B.: Convex Polytopes. Wiley, New York (1967)
Gritzmann, P., Klee, V.: On the complexity of some basic problems in computational convexity II, volume and mixed volumes. In: Polytopes: Abstract, Convex and Computational. Nato Advanced Study Institute. Serie C, Mathematical and Physical Sciences, pp. 379–466. Kluwer Academic, Dordrecht (1995)
Horst, R., Thoai, N.V., de Vries, J.: On finding new vertices and redundant f constraints in cutting plane algorithms for global optimization. Oper. Res. Lett. 7, 85–90 (1988)
Chen, P.C., Hansen, P., Jaumard, B.: On-line and off-line vertex enumeration by adjacency lists. Oper. Res. Lett. 10, 403–409 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Leitmann.
Rights and permissions
About this article
Cite this article
Horst, R. Bisecton by Global Optimization Revisited. J Optim Theory Appl 144, 501–510 (2010). https://doi.org/10.1007/s10957-009-9610-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-009-9610-8