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Improved lower bounds for optimization problems related to atom clusters 255Table 1. Lower bounds for the minimal distances in optimal three dimensional Morse clusters for different ρ. Thelast column contains linear lower bounds for the optimal values.ρ Locatelli-Schoen [4] Vinkó [5] present work lower bound on the optimal value6 0.114 0.499 0.559 −81.23699155n7 0.376 0.611 0.652 −51.83256650n8 0.468 0.680 0.710 −40.34618808n9 0.528 0.727 0.752 −33.74384634n10 0.574 0.762 0.782 −29.76671162n11 0.613 0.789 0.807 −26.58857202n12 0.644 0.810 0.826 −24.62136302n13 0.672 0.828 0.841 −23.40692214n14 0.695 0.842 0.854 −22.11827825n15 0.715 0.855 0.865 −21.19287270nresults are achieved using the fact that by [4] q must be greater than the second column inTable 1.5. Usefulness of the resultsThe information on the minimal interatomic distance can be appliedin Branch-and-Bound methods as an accelerating tool;in a starting point generator used in incomplete or asymptotically complete global solvers.For instance, in [3] this kind of information is used to improve the performace of the proposedsolving technique;and –as it is proved in [7]– one can construct efficient data structure to accelerate thecomputation of the potential function. Surprisingly, the value of the potential function(1) can be computed in O(n) time (with a naive method we have O(n 2 ) time complexity).The lower bound for the global minimum can be used in Branch-and-Bound methods as acut-off test.References[1] X. Blanc. Lower bounds for the interatomic distance in Lennard-Jones clusters. Computational Optimization andApplications 29:5-12,2004.[2] J.P.K. Doye, R.H. Leary, M. Locatelli and F. Schoen. The global optimization of Morse clusters by potentialenergy transformations. INFORMS Journal On Computing, 16:371-379, 2004.[3] M. Locatelli and F. Schoen. Fast global optimization of difficult Lennard-Jones clusters Computational Optimizationand Applications, 21:55-70, 2002.[4] M. Locatelli and F. Schoen. Minimal interatomic distance in Morse-clusters. Journal of Global Optimization22:175-190, 2002.[5] T. Vinkó. Minimal inter-particle distance in atom clusters. To appear in Acta Cybernetica. 2005.[6] G.L. Xue. Minimum inter-particle distance at global minimizers of Lennard-Jones clusters. Journal of GlobalOptimization, 11:83-90, 1997.[7] G.L. Xue. An O(n) time hierarchical tree algorithm for computing force field in n-body simulations. TheoreticalComputer Science, 197:157-169, 1998.