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Diagonal global search based on a set of possible Lipschitz constants 223Table 5.Improvement obtained by the new algorithm with respect to DIRECT and DIRECTl.N ∆ Class GKLS Shifted GKLSr ρ DIRECT/New DIRECTl/New DIRECT/New DIRECTl/New2 10 −4 0.90 0.20 2.88 5.75 2.70 3.892 10 −4 0.90 0.10 1.77 1.89 1.68 1.443 10 −6 0.66 0.20 4.99 5.31 3.29 5.343 10 −6 0.90 0.20 >166.50 4.87 3.53 6.864 10 −6 0.66 0.20 >68.87 8.18 2.55 6.614 10 −6 0.90 0.20 >23.45 6.75 7.67 8.565 10 −7 0.66 0.30 >29.82 5.31 6.70 11.135 10 −7 0.66 0.20 >10.67 >10.67 5.83 >12.82functions of a particular class (columns “50%”) and for all 100 function of the class (columns“100%”). The notation “> 1 000 000 (k)” means that after 1 000 000 trials the method underconsideration was not able to solve k problems.In order to avoid a redundant accumulation of trial points generated by DIRECT near theparaboloid vertex (with the function value equal to 0) and to put DIRECT and DIRECTl in amore advantageous situation, we shifted all generated functions, adding to their values theconstant 2. In such a way, the value of each function at the paraboloid vertex became equalto 2.0 and the global minimum value f ∗ was increased by 2, i.e., became equal to 1.0. Results ofnumerical experiments with shifted GKLS classes (defined in the rest by the same parameters)are reported in Tables 3 and 4.Note that on a half of test functions from each class (which were simple for each methodwith respect to the other functions of the class) the new algorithm manifested a good performancewith respect to DIRECT and DIRECTl in terms of the number of generated trialpoints (Tables 1, 3). When all functions were taken in consideration, the number of trials producedby the new algorithm was significantly fewer in comparison with two other methods(see columns “100%” of Tables 1, 3), providing at the same time a good examination of theadmissible region (see Tables 2, 4).Table 5 summarizes (based on the data from Tables 1 – 4) results of numerical experimentsperformed on 1600 test functions from GKLS and shifted GKLS continuously differentiableclasses. It represents the ratio between the maximal number of trials performed by DIRECTand DIRECTl with respect to the corresponding number of trials performed by the new algorithm.As it can be seen from Tables 1 – 5, the new method demonstrates a quite satisfactory performancewith respect to DIRECT [6] and DIRECTl [1, 2] when multidimensional functionswith a complex structure are minimized.References[1] J. M. Gablonsky. Modifications of the DIRECT algorithm. PhD thesis, North Carolina State University, Raleigh,North Carolina, 2001.[2] J. M. Gablonsky and C. T. Kelley. A locally-biased form of the DIRECT algorithm. J. Global Optim., 21(1):27–37, 2001.[3] M. Gaviano, D. Lera, D. E. Kvasov, and Ya. D. Sergeyev. Algorithm 829: Software for generation of classesof test functions with known local and global minima for global optimization. ACM Trans. Math. Software,29(4):469–480, 2003.[4] R. Horst and P. M. Pardalos, editors. Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht,1995.