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76 Leocadio G. Casado, Eligius M.T. Hendrix, and Inmaculada Garcíalowest integer greater than or equal to x. Given the number of edges per simplex n(n − 1)/2,the maximum depth of the search tree is K × n(n − 1)/2. The final level is not stored, as thesimplices don’t pass the size test. An overestimate for the worst case number of simplices onthe list is 2 K×n(n−1)/2−1 , where K = ⌈(−lnɛ/ln(2))⌉. This analysis provides a guarantee thatthe algorithm is finite given an accuracy. Looking more realistically, this upper bound of anupper bound also leads to dispare; for ɛ = 1%, it gives a bound of the order of 10 6 for n = 3and 10 44 for n = 7. This does not sound very encouraging nor realistic.Let us now consider a lower bound on the worst case behaviour. Consider again the regulargrid in Figure 2. Suppose an algorithm would generate with an accuracy of ɛ = 1M−1 allcorresponding simplices of the regular mesh. We know that bisection has to generate more inthe worst case. How many simplices would the algorithm generate? The number of simplicesin the regular grid is(M − 1) n−1 (4)Formula (4) can be derived from volume considerations. The unit simplex represents a volumein n − 1 dimensional space proportional to Size n−1 . As the size of a simplices within theregular grid has a size of1M−1 times the unit simplex size, its volume is (1M−1) n−1that of theunit simplex. Also the number (4) turns out big; for ɛ = 1%, it gives 10 4 for n = 3 and 10 12 forn = 7. We will observe in the experiments that practically the number is much lower. The realsuccess of Branch-and-Bound depends on how good parts of the tree can be pruned. Searchstrategies on deep or wide search determine the final result.4. SummaryThis extended abstract describes the quadratic blending problem with additional complicationsof semi-continuity and bi-objectivity. The ingredients are sketched of a Branch-and-Bound approach under investigation. Some convergence analyses based on the benchmark ofgrid search are given. In the full paper empirical results on cases derived from industry arereported.References[1] Eligius M. T. Hendrix and Janos D. Pinter. An application of Lipschitzian global optimization to productdesign. J. Global Optim., 1:389–401, 1991.[2] R. Horst, P.M. Pardalos, and N.V. Thoai, editors. Introduction to Global Optimization, volume 3 of NoncovexOptimization and its Applications. Kluwer Academic Publishers, Dordrecht, Holland, 1995.[3] H. P. Williams. Model Building in Mathematical Programming. Wiley & Sons, Chichester, 1993.