Abstract
During the last 40 years, simplicial partitioning has been shown to be highly useful, including in the field of nonlinear optimization, specifically global optimization. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on this exhaustivity which seem at first glance to be true (two of which have been stated as true in published articles). However, we will provide counter-examples to these conjectures. We also provide a new simplicial partitioning scheme, which provides a lot of freedom, whilst guaranteeing exhaustivity.
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Bomze, I.M., Eichfelder, G.: Copositivity detection by difference-of-convex decomposition and \(\omega \)-subdivision. To appear in: Math. Program. (2012). doi:10.1007/s10107-012-0543-x
Bundfuss, S., Dür, M.: Algorithmic copositivity detection by simplicial partition. Linear Algebra Appl. 428(7), 1511–1523 (2008). doi:10.1016/j.laa.2007.09.035
Hendrix, E.M., Casado, L.G., García, I.: The semi-continuous quadratic mixture design problem: description and branch-and-bound approach. Eur. J. Oper. Res. 191(3), 803–815 (2008). doi:10.1016/j.ejor.2007.01.056
Horst, R.: An algorithm for nonconvex programming problems. Math. Program. 10(1), 312–321 (1976). doi:10.1007/BF01580678
Horst, R.: On generalized bisection of \(n\)-simplices. Math. Comput. 66(218), 691–698 (1997). doi: 10.1090/S0025-5718-97-00809-0
Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Springer, Berlin (2000)
Kearfott, B.: A proof of convergence and an error bound for the method of bisection in \({\mathbb{R}}^n\). Math. Comput. 32(144), 1147–1153 (1978). doi: 10.2307/2006341
Koepf, W.: Hypergeometric summation: an algorithmic approach to summation and special function identities. Friedrich Vieweg & Sohn Verlagsgesellschaft mbH (1998)
Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39(2), 117–129 (1987). doi:10.1007/BF02592948
Tuy, H.: Effect of the subdivision strategy on convergence and efficiency of some global optimization algorithms. J. Glob. Optim. 1(1), 23–36 (1991). doi:10.1007/BF00120663
Tuy, H.: Normal conical algorithm for concave minimization over polytopes. Math. Program. 51(1), 229–245 (1991). doi:10.1007/BF01586935
Tuy, H., Horst, R.: Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and d.c. optimization problems. Math. Program. 41(1), 161–183 (1988). doi:10.1007/BF01580762
Žilinskas, A., Žilinskas, J.: Global optimization based on a statistical model and simplicial partitioning. Comput. Math. Appl. 44(7), 957–967 (2002). doi:10.1016/S0898-1221(02)00206-7
Žilinskas, J.: Branch and bound with simplicial partitions for global optimization. Math. Model. Anal. 13(1), 145–159 (2008). doi:10.3846/1392-6292.2008.13.145-159
Žilinskas, J., Dür, M.: Depth-first simplicial partition for copositivity detection, with an application to maxclique. Optim. Methods Softw. 26(3), 499–510 (2011). doi:10.1080/10556788.2010.544310
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The author wishes to thank Mirjam Dür and the anonymous referees for their useful comments with regards to this paper.
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Dickinson, P.J.C. On the exhaustivity of simplicial partitioning. J Glob Optim 58, 189–203 (2014). https://doi.org/10.1007/s10898-013-0040-7
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DOI: https://doi.org/10.1007/s10898-013-0040-7