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246 Boglárka Tóth and L.G. CasadoOne can see, that the nearer a sf jto 1/n, the grater the volume is. It also implies thatknowing the vector (sf 1, . . . , sf n) we can choose the best directions to shift.Let us note, that if we have the volume for the SPB shifted in the dimensions j 1 , . . . , j k−1 ∈{1, . . . , n}, and we want to shift it in one more dimension, j k ∈ {1, . . . , n} too, the new volumecan be computed from the known V (SP B j1 ,...,j k−1) in the following way:( ) ( )n − k + 1n−kn−kosfj1,...,jV (SP B j1 ,...,j k) = sfk−1− sf jk n − k + 1jkV (SP B j1 ,...,jn − kosf j1 ,...,j k−1osf k−1)j1 ,...,j k−1Similarly to Theorem 4 it can be shown that if CP B is not inside OB we can always showan SP B with larger volume inside OB. Although in the next theorem we state a bit more.Theorem 12. Suppose CP B ⊄ OB in the dimensions j 1 , . . . , j k ∈ {1, . . . , n}, (k < n), i.e. ob U i cpb L i for all i ∈ {j 1 , . . . , j k }, but not for i ∈ {1, . . . , n} \ {j 1 , . . . , j k }. ThenSP B j1 ,...,j kis such that V (SP B j1 ,...,j k) > V (CP B ∩ OB), and V (SP B j1 ,...,j k) > V (SP B j1 ,...,j l∩OB), l < k.5. Results and conclusionsThe presented pruning method was included in a general Interval Global Optimization algorithmwhich use centered form and naiv interval arithmetic as inclusion functions and monotonicitytest, for some hard test problems. Table 1 shows the effort of the algorithm withoutand with the new pruning method which let to cut the pruneable box with the best pruningrate = volume ratio/new boxes rate. The smallest allowed pruning rate was 0.035. One can see,that the pruning method works for multi-dimensional cases, and it is worth to incorporate itto Interval Global Optimization Algorithms.Table 1.Numerical results for the use of the new pruning method compared to the general B&B algorithm.Function name n ε Effort Effort With Pruning SpeedUpRatz 3 10 −3 633508 535196 1.18Kowalik 4 10 −3 4883230 2125330 2.30EX2 5 10 −3 3448563 1970996 1.75Ratz 5 10 −3 1201476 1035922 1.16Ratz 7 10 −3 1900512 1667050 1.14Ratz 9 10 −3 3183190 2435704 1.31Griewank 10 10 −3 1551366 1551366 1.00Rastrigin 10 10 −3 1551366 1593770 0.97Rosenbrock 10 10 −3 510041 335433 1.52#F : number of function evaluation, #G: number of gradient evaluation, Effort= #F + n#GReferences[1] E. Baumann. Optimal centered forms. BIT, 28(1):80–87, 1988.[2] R.E. Moore. Interval analysis. Prentice-Hall, New Jersey, (USA), 1966.[3] D. Ratz. Automatic Slope Computation and its Application in Nonsmooth Global Optimization. Shaker Verlag,Aachen, Germany, 1998.