Abstract
In this paper, the use of non-optimality spheres in a simplicial branch and bound (B&B) algorithm is investigated. In this context, some considerations regarding the use of bisection on the longest edge in relation with ideas of Reiner Horst are reminded. Three arguments highlight the merits of bisection of simplicial subsets in B&B schemes.
This paper has been supported by The Spanish Ministry of Science and Innovation (project TIN2008-01117) and Junta de Andalucía (P11-TIC-7176), in part financed by the European Regional Development Fund (ERDF) and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through PEst-OE/MAT/UI0297/2011 (CMA). Eligius M.T. Hendrix is a fellow of the Spanish “Ramon y Cajal” contract program, co-financed by the European Social Fund.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Baritompa, W.P.: Customizing methods for global optimization, a geometric viewpoint. Journal of Global Optimization 3, 193–212 (1993)
Breiman, L., Cutler, A.: A deterministic algorithm for global optimization. Mathematical Programming 58, 179–199 (1993)
Casado, L.G., García, I., Tóth, B.G., Hendrix, E.M.T.: On determining the cover of a simplex by spheres centered at its vertices. Journal of Global Optimization 50, 654–655 (2011)
Casado, L.G., Hendrix, E.M.T., García, I.: Infeasibility spheres for finding robust solutions of blending problems with quadratic constraints. Journal of Global Optimization 39, 557–593 (2007)
Danilin, Y., Piyavski, S.A.: An algorithm for finding the absolute minimum. Theory of Optimal Decisions 2, 25–37 (1967) (in Russian)
Evtushenko, Y., Posypkin, M.: Coverings for global optimization of partial-integer nonlinear problems. Doklady Mathematics 83, 1–4 (2011)
Horst, R.: On generalized bisection of n-simplices. Mathematics of Computation 66(218), 691–698 (1997)
Horst, R.: Bisection by global optimization revisited. Journal of Optimization Theory and Applications 144, 501–510 (2010)
Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, Nonconvex Optimization and its Applications, vol. 3. Kluwer Academic Publishers, Dordrecht (1995)
Horst, R., Tuy, H.: On the convergence of global methods in multiextremal optimization. Journal of Optimization Theory and Applications 54, 253–271 (1987)
Horst, R., Tuy, H.: Global Optimization (Deterministic Approaches). Springer, Berlin (1990)
Locatelli, M., Raber, U.: On convergence of the simplicial branch-and-bound algorithm based on ω-subdivisions. J. Optim. Theory Appl. 107, 69–79 (2000)
Mladineo, R.H.: An algorithm for finding the global maximum of a multimodal multivariate function. Mathematical Programming 34, 188–200 (1986)
Paulavičius, R., Žilinskas, J., Grothey, A.: Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of lipschitz bounds. Optimization Letters 4, 173–183 (2010)
Piyavski, S.A.: An algorithm for finding the absolute extremum of a function. USSR Computational Mathematics and Mathematical Physics 12, 57–67 (1972) (in Russian)
Raber, U.: A simplicial branch-and-bound method for solving nonconvex all-quadratic programs. Journal of Global Optimization 13, 417–432 (1998)
Raber, U.: Nonconvex All-Quadratic Global Optimization Problems: Solution Methods, Application and Related Topics. Ph.D. thesis, Trier University (1999)
Shubert, B.O.: A sequential method seeking the global maximum of a function. SIAM Journal of Numerical Analysis 9, 379–388 (1972)
Zilinskas, A., Clausen, J.: Subdivision, sampling, and initialization strategies for simplicial branch and bound in global optimization. International Journal of Computers and Mathematics with Applications 44, 943–955 (2002)
Zilinskas, A., Zilinskas, J.: Global optimization based on a statistical model and simplicial partitioning. International Journal of Computers and Mathematics with Applications 44, 957–967 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hendrix, E.M.T., Casado, L.G., Amaral, P. (2012). Global Optimization Simplex Bisection Revisited Based on Considerations by Reiner Horst. In: Murgante, B., et al. Computational Science and Its Applications – ICCSA 2012. ICCSA 2012. Lecture Notes in Computer Science, vol 7335. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31137-6_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-31137-6_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31136-9
Online ISBN: 978-3-642-31137-6
eBook Packages: Computer ScienceComputer Science (R0)