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Branch-and-Bound for the semi-continous quadratic mixture design problem (SCQMDP) 75Figure 2. 2-dim projection of regular grid over the unit simplex, M=5as sketched in Table 1. This means that the number of points, although very high, is not evenexponential in the dimension. For the example in Figure 2, n = 3 and m = 5, we have 15 gridpoints and an "accuracy" of α = 0.25.Table 1.Number of regular grid points on the unit simplexn=2 n=3 n=4 n=5 n=6 n=7M=11 11 66 286 1001 3003 8008M=101 101 5151 176851 4.6 10 6 9.7 10 7 1.7 10 9Notice that the concept of Branch-and-Bound is not to generate all these points, as we canthrow out parts of the area where the optima cannot be found. However, if one studies thebisection process, we arrive for the same example at more points as illustrated by Figure 3.Figure 3. Bisection process 2-dim projection, ɛ ≥ 0.25As such this is not a disaster. The main question is what happens with the storage of subsets;the number of stored simplices should not grow out of hand. Before dealing with that, letus remark with respect to Figure 3, that the picture does not appear straightforward. In theimplementation of the bisection, a choice rule is required that determines in case of equal sizedlargest edges, which one is to be bisected first. In our implementations we did so on the basisof the co-ordinates. To ensure reproducibility of results, it should not be chosen at random.The number of simplices that is generated (and stored) in the worst case, depends on manyaspects. We could derive an upper bound and a lower bound on the worst case performance.In the worst case, rules lead to splitting and storing simplices that have a size slightly largerthan ɛ. At every bisection over a midpoint, one edge is halved and several are getting shorter.This means that after going n(n − 1)/2 deeper in the search tree, at least all edges have beenhalved and the size of the simplex is less than half its original size. The maximum number Kof halving the simplices is given by 1/2 K ≤ ɛ, such that K = ⌈(−lnɛ/ln(2))⌉, where ⌈x⌉ is the