### Table 3. Test Functions for Global Optimization Function Constraints

1995

"... In PAGE 8: ... 123 { 141. Berlin: Springer{Verlag 1995 For the test functions of Table3 we get the results shown in Figure 5 (Schwe- fel apos;s function F7 is normalized to make a log-plot possible) which con rm the relation (6). For these function we observe di erent regions of convergence.... In PAGE 15: ...able 5. lt;feval gt; vs. n for Function F6 (left) with f 9 10?1, I = 1:4, Rm = 0:1 , Rmin = 10?1 , pm = 1=n , bm = 2, 20 runs, and for Function F7 (right) with f fopt + 5 10?4 jfoptj , I = 1:4, Rm = 0:75 ,Rmin = 10?4 , pm = 1=n , bm = 2, 20 runs, lt;feval gt; vs. n for Function F8 (left) with f 10?3, I = 1:4, Rm = 0:1 , Rmin = 10?6 , pm = 1=n , bm = 2, 20 runs, and for Function F9 (right) with f 10?3 , I = 1:4, Rm = 0:1 ,Rmin = 10?4, pm = 1=n , bm = 2, 20 runs, BGA data from [10] Rastrigin apos;s Function F6 Schwefel apos;s Function F7 n EASY BGA n EASY BGA N lt;feval gt; N lt;feval gt; N lt;feval gt; N lt;feval gt; 20 20 6098 20 3608 20 20 10987 500 16100 100 20 45118 20 25040 100 20 101458 1000 92000 200 20 98047 20 52948 200 20 241478 2000 248000 400 20 243068 20 112634 400 20 430084 4000 699803 1000 20 574561 20 337570 1000 20 1067221 || ||| Griewangk apos;s Function F8 Ackley apos;s Function F9 n EASY BGA n EASY BGA N lt;feval gt; N lt;feval gt; N lt;feval gt; N lt;feval gt; 20 500 26700 500 66000 30 20 13997 20 19420 100 500 77250 500 361722 100 20 57628 20 53860 200 500 128875 500 748300 200 20 122347 20 107800 400 500 229750 500 1630000 400 20 262606 20 220820 1000 500 563350 | ||| 1000 20 686614 20 548306 The test functions given in Table3 are very popular in the literature on global optimization. The main results of this paper are summarized in Table 5 with n the number of variables, N the population size, and feval the average number of function evaluations for 20 runs.... ..."

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### Table 4 Globalized functions for the output variables of (1.13).

1996

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### Table 4 Globalized functions for the output variables of (1.13).

### Table 1: Global Preference Function Scale

1997

"... In PAGE 4: ... Based on discussions with experts and work to classify previous matches into various sets of linguistic terms we found that there was enough precision in our evaluation of the similarity of the attributes to have four linguistic terms. Table1 shows the linguistic terms and the numeric similarity score that corresponds to each term. Table 1: Global Preference Function Scale... In PAGE 16: ... Based on discussions with experts and work to classify previous matches into various sets of linguistic terms we found that there was enough precision in our evaluation of the similarity of the attributes to have four linguistic terms. Table1 shows the linguistic terms and the numeric similarity score that corresponds to each term. Table 1: Global Preference Function Scale Fuzzy Rating Maximum Score Minimum Score Excellent 1 0.... In PAGE 16: ... Table 1 shows the linguistic terms and the numeric similarity score that corresponds to each term. Table1 : Global Preference Function Scale Fuzzy Rating Maximum Score Minimum Score Excellent 1 0.95 Good 0.... ..."

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### Table 1: Global Preference Function Scale

1997

"... In PAGE 4: ... Based on discussions with experts and work to classify previous matches into various sets of linguistic terms we found that there was enough precision in our evaluation of the similarity of the attributes to have four linguistic terms. Table1 shows the linguistic terms and the numeric similarity score that corresponds to each term. Table 1: Global Preference Function Scale Fuzzy Rating Maximum Score Minimum Score Excellent 1 0.... In PAGE 4: ... Table 1 shows the linguistic terms and the numeric similarity score that corresponds to each term. Table1 : Global Preference Function Scale Fuzzy Rating Maximum Score Minimum Score Excellent 1 0.95 Good 0.... In PAGE 16: ... Based on discussions with experts and work to classify previous matches into various sets of linguistic terms we found that there was enough precision in our evaluation of the similarity of the attributes to have four linguistic terms. Table1 shows the linguistic terms and the numeric similarity score that corresponds to each term. Table 1: Global Preference Function Scale... ..."

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### Table 1: Global and local function approximation methods.

1996

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### Table 6: Global approximations of the E-function.

1998

"... In PAGE 34: ... Below we say that local approximation Ai is included in a global approximation if at least one of the masks of this approximation is non-zero. Table6 lists all the useful global approximations of the E-function, according to which local approximations are included in them. We remark that a certain subset of the local approximations can give rise to many different global approximations, depending on the setting of the relevant masks.... In PAGE 34: ... Example 1. Consider an approximation of the E-function which only uses local approximations A1; A2 and A3 (Line 3 in Table6 ). A conceivable way to devise such approximation is to assign non-zero values only to the masks X1; X2; X3; X4; X6; X9; X10 and X11 (and possibly also to X5), in such a way that X4 = X1 lt; 13, X3 = X9 and X10 = X6 lt; 5.... In PAGE 35: ... Example 2. Consider an approximation of the E-function which uses local approximations A1, A2, A4, A5, A6, A7 (Line 2 in Table6 ). Again, it is conceivable that such an approximation can set values for the involved masks so that X4 = X1 lt; 13, X6 (X13 lt; 5) (X16 lt; 10) (X19 lt; 10) = 0, and X3 = X7, in which case the resulting global approximation is of the form (X2 k1) (X5 k2) (X20 L) (6) A similar global approximation can be obtained from Line 12 in Table 6, except that in that case we also get X2 = X5 = 0.... In PAGE 35: ... Consider an approximation of the E-function which uses local approximations A1, A2, A4, A5, A6, A7 (Line 2 in Table 6). Again, it is conceivable that such an approximation can set values for the involved masks so that X4 = X1 lt; 13, X6 (X13 lt; 5) (X16 lt; 10) (X19 lt; 10) = 0, and X3 = X7, in which case the resulting global approximation is of the form (X2 k1) (X5 k2) (X20 L) (6) A similar global approximation can be obtained from Line 12 in Table6 , except that in that case we also get X2 = X5 = 0. As in the previous example, the problem here too is to assign values to the masks so as to get an approximation with non-zero bias.... ..."

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### Table 1. Global Radial Basis Functions

2006

"... In PAGE 2: ... Here (k k), where k k is the Euclidean norm, is any sufficiently differentiable semi-positive radial basis function. Substituting (4) in (1) and applying the boundary conditions (2), we obtain N X j=1 d j dt (kx xjk) + jL (kx xjk) = 0; (5) N X j=1 j (kx xjk) = g(xi; t); (6) where L denote the application of the spatial derivatives on the kernel function (see Table1 ), with the indices i = 1; : : : ; NI for (5) and i = NI + 1; NI + 2; : : : ; N for (6). The time derivative is approximate by a first-order time dif- ference scheme, obtaining d~ u(x; t) dt = N X j=1 1 t( t+ t j t j) (kx xjk); (7) where t is the time step.... ..."

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### Table 1. Global Radial Basis Functions

2006

"... In PAGE 2: ... The linear system is solved by LU factorization. The most widely used radial basis functions are shown in Table1 , with r = kxi xjk. In our numerical examples we have used the Thin-Plate Splines (TPS) with m = 4, which avoid the complexity introduced for the selection of the shape parameter c used in MQ, IMQ and GA.... ..."

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