Multidimensional Benchmarks Results

This page shows the results obtained by applying a number of Global optimization algorithms to the entire benchmark suite of N-D optimization problems, together with some statistics on the algorithm performances.

test_functions Multi-Dimensional (N-D) Test Functions

The following table shows the overall success of all Global Optimization algorithms, considering for every benchmark function 100 random starting points.

So, for example, AMPGO was able to solve, on average, 79.0% of all the test functions for all the 100 random starting points using, on average, 611 functions evaluations.

Optimization algorithms performances (N-dimensional)
Optimization Method Overall Success (%) Functions Evaluations
AMPGO 79.027 611
ASA 58.457 3566
BasinHopping 49.478 1220
CMA-ES 56.783 605
CRS2 57.332 966
DE 25.533 1429
DIRECT 55.027 563
Firefly 53.565 1462
Galileo 4.967 127
MLSL 20.147 3093
PSWARM 52.995 1165
SIMANN 6.446 1390

These results are also depicted in the next figure, which clearly shows that AMPGO is the better-performing optimization algorithms as far as the current benchmark is considered.

AMPGO N-D results

AMPGO Optimization algorithms performances (N-dimensional)


It is also interesting to analyze the success of an optimization algorithm based on the fraction (or percentage) of problems solved given a fixed number of allowed function evaluations, let’s say from 100 to 2000. The next figure presents such analysis, and it is clear that, on average, AMPGO outperforms all the other optimization methods using only 200 function evaluations (against the 2000 allowed for all the others).

AMPGO N-D results

AMPGO Percentage of problems solved given a fixed number of function evaluations (N-dimensional)


After the following comment from Geoffrey Oxberry in the scipy mailing list:

Note

One of the typical plots used to assess performance is a “performance profile”, which was defined in “Benchmarking Optimization Software with Performance Profiles” by Dolan and More (http://arxiv.org/pdf/cs/0102001.pdf). I didn’t see any plots in this format. Do you plan on presenting performance data in this manner? The solved problems versus function evaluations looks pretty close to this sort of presentation. This sort of format also avoids some of the pitfalls mentioned on your site re: performance comparisons.

I converted the Matlab code used by Dolan and More to Python and produced the following output:

AMPGO N-D results

AMPGO Performance profiles for the various optimization algorithms (N-dimensional)


The following table is a split-by-benchmark function of the first table, showing the percentage of successful optimizations per benchmark, considering 100 random starting points.

Optimization algorithms performances (N-dimensional)
Function Name N AMPGO ASA BasinHopping CMA-ES CRS2 DE DIRECT Firefly Galileo MLSL PSWARM SIMANN
Ackley 2 43 99 1 18 85 0 100 0 0 0 1 0
Adjiman 2 100 100 100 99 97 100 0 100 0 0 100 0
Alpine01 2 100 100 0 95 67 0 100 1 0 100 65 0
Alpine02 2 98 100 31 43 88 100 100 97 0 0 69 74
AMGM 2 100 100 100 100 57 100 0 100 60 0 90 0
BartelsConn 2 100 100 0 100 92 0 0 13 0 0 34 0
Beale 2 100 3 100 61 63 0 100 78 0 0 76 0
Bird 2 100 100 44 61 62 100 100 70 0 0 72 0
Bohachevsky 2 100 100 100 88 91 0 100 95 0 0 93 0
BoxBetts 3 100 0 92 91 89 0 0 86 0 0 8 0
Branin01 2 100 100 100 100 81 0 100 100 0 0 70 0
Branin02 2 100 18 24 7 18 0 100 6 0 0 6 0
Brent 2 100 100 100 100 35 0 100 100 0 0 100 0
Brown 2 100 96 100 100 88 0 100 100 0 0 100 0
Bukin02 2 100 100 100 100 100 100 100 100 100 0 100 0
Bukin04 2 100 100 0 99 85 100 0 61 0 100 100 0
Bukin06 2 94 0 0 0 0 0 0 0 0 0 0 0
CarromTable 2 100 100 41 13 48 100 100 93 65 0 99 0
Chichinadze 2 71 100 18 6 49 0 54 15 0 0 65 0
Cigar 2 100 98 0 100 71 0 0 0 0 100 27 0
Cola 17 58 0 0 0 0 0 0 0 0 0 0 0
Colville 4 100 0 0 100 0 100 100 100 12 100 87 1
Corana 4 0 100 55 5 0 0 0 4 0 0 62 0
CosineMixture 2 100 100 100 100 0 100 100 100 100 0 99 0
CrossInTray 2 79 100 4 13 97 0 100 100 0 0 100 0
CrossLegTable 2 0 0 0 0 0 0 0 0 0 0 0 9
CrownedCross 2 3 0 0 0 0 0 0 0 0 0 0 0
Csendes 2 100 100 100 100 100 0 100 100 13 0 100 0
Cube 2 100 0 0 96 65 51 0 19 0 54 50 0
Damavandi 2 18 0 0 0 0 0 0 0 0 0 0 0
Deb01 2 100 100 100 100 42 100 0 100 0 100 30 0
Deb02 2 100 100 100 93 95 0 100 98 0 100 23 0
Decanomial 2 93 35 0 98 97 0 0 20 0 38 99 0
Deceptive 2 31 98 0 0 6 0 0 0 0 100 2 0
DeckkersAarts 2 100 100 99 100 95 100 100 83 0 0 100 0
DeflectedCorrugatedSpring 2 96 0 100 3 38 100 0 65 0 0 100 0
DeVilliersGlasser01 4 100 0 7 4 0 0 0 0 0 100 0 0
DeVilliersGlasser02 5 1 0 0 0 0 0 0 0 0 0 0 0
DixonPrice 2 100 97 100 100 86 56 100 100 0 41 100 0
Dolan 5 100 0 0 83 0 0 100 4 0 100 20 0
DropWave 2 34 20 93 0 39 0 0 57 0 0 12 0
Easom 2 5 74 0 1 26 0 0 0 0 62 0 0
EggCrate 2 94 100 31 17 77 0 100 87 0 47 100 0
EggHolder 2 59 2 2 0 16 0 0 7 0 0 25 100
ElAttarVidyasagarDutta 2 100 44 62 57 70 0 100 1 0 0 57 0
Exp2 2 100 98 10 3 28 0 0 1 0 0 47 0
Exponential 2 100 100 100 99 95 47 100 100 0 0 100 0
FreudensteinRoth 2 100 58 25 34 57 0 0 28 0 100 97 0
Gear 4 44 100 3 98 96 100 0 100 2 0 83 0
Giunta 2 100 100 69 59 91 0 100 93 0 0 100 0
GoldsteinPrice 2 100 74 94 75 83 0 100 97 0 0 98 0
Griewank 2 3 17 0 0 51 0 0 3 0 0 0 0
Gulf 3 59 0 3 0 44 0 0 0 0 0 0 0
Hansen 2 80 56 80 9 41 52 51 91 0 0 25 0
Hartmann3 3 100 0 100 95 86 0 0 26 0 0 100 0
Hartmann6 6 100 0 20 68 0 0 100 54 0 0 70 0
HelicalValley 3 100 0 100 100 10 0 52 0 0 81 2 0
HimmelBlau 2 100 0 100 100 87 0 100 100 0 0 56 0
HolderTable 2 100 100 52 18 92 100 100 95 95 0 100 0
Holzman 3 100 100 80 35 0 100 0 100 0 100 0 100
Hosaki 2 100 100 87 82 95 100 100 99 0 0 100 0
Infinity 2 100 100 100 100 98 0 100 100 8 0 100 0
JennrichSampson 2 100 2 0 100 78 0 50 61 0 0 62 0
Judge 2 100 8 100 82 84 0 41 100 0 0 92 0
Katsuura 2 100 92 100 98 0 0 100 100 0 0 100 0
Keane 2 100 100 100 100 65 100 0 100 100 100 8 0
Kowalik 4 92 0 4 18 1 0 0 0 0 0 0 0
Langermann 2 97 70 71 12 67 100 0 62 0 0 78 0
LennardJones 6 100 0 90 79 3 0 100 100 0 100 58 0
Leon 2 100 0 0 100 58 0 56 44 0 0 0 0
Levy03 2 100 100 96 32 96 58 0 100 1 56 100 0
Levy05 2 26 50 74 8 51 100 43 81 0 0 62 0
Levy13 2 100 100 100 63 81 58 0 100 0 47 74 0
Matyas 2 100 31 100 100 81 44 100 100 0 0 100 0
McCormick 2 100 100 52 61 65 0 100 99 0 0 100 0
Michalewicz 2 100 100 100 43 82 100 100 90 0 0 100 36
MieleCantrell 4 93 1 97 99 97 0 50 51 0 0 29 0
Mishra01 2 100 100 100 98 0 100 100 100 0 0 4 0
Mishra02 2 100 100 100 100 0 100 100 100 0 0 100 0
Mishra03 2 43 7 0 2 15 0 50 0 0 0 20 93
Mishra04 2 29 3 0 0 14 0 0 2 0 0 10 91
Mishra05 2 97 78 88 10 97 0 44 21 1 0 60 0
Mishra06 2 100 54 36 38 81 47 100 91 0 0 89 0
Mishra07 2 100 100 100 100 93 53 100 100 2 100 54 0
Mishra08 2 94 38 0 94 90 0 0 18 0 91 96 0
Mishra09 3 100 100 0 100 52 0 100 41 0 61 19 0
Mishra10 2 0 99 20 51 0 100 0 99 37 0 89 78
Mishra11 2 100 100 100 100 91 53 100 100 3 47 75 5
MultiModal 2 100 100 0 100 98 0 100 97 0 100 100 0
NeedleEye 2 100 100 0 100 0 100 0 37 0 0 100 0
NewFunction01 2 60 7 0 4 44 0 0 4 0 0 18 35
NewFunction02 2 42 1 0 1 14 0 0 1 0 0 3 36
NewFunction03 2 100 83 100 28 99 100 57 88 18 0 99 0
OddSquare 2 6 18 2 4 93 100 0 100 0 0 73 2
Parsopoulos 2 100 100 89 93 55 0 100 100 0 0 34 0
Pathological 2 30 15 8 0 0 0 100 0 0 0 93 0
Paviani 10 100 0 100 100 0 0 0 0 0 0 0 0
Penalty01 2 100 99 38 88 89 0 100 100 0 0 61 0
Penalty02 2 52 43 100 37 84 0 0 53 0 0 14 0
PenHolder 2 100 100 19 27 99 100 100 98 89 0 100 100
PermFunction01 2 100 0 100 100 79 0 100 100 0 100 15 0
PermFunction02 2 100 0 100 100 85 0 100 100 0 0 97 0
Pinter 2 96 82 97 31 79 0 100 98 0 52 53 0
Plateau 2 43 100 53 81 0 100 0 100 90 0 100 100
Powell 4 100 0 0 100 43 0 0 2 0 0 2 0
PowerSum 4 92 0 0 13 0 0 0 0 0 0 0 0
Price01 2 100 100 100 98 79 0 100 6 0 55 100 0
Price02 2 65 43 1 3 49 0 0 36 0 0 27 0
Price03 2 100 5 55 57 35 0 100 7 0 0 58 0
Price04 2 100 18 4 100 82 42 44 65 0 74 23 0
Qing 2 100 100 100 99 79 0 100 2 0 0 100 0
Quadratic 2 100 100 100 100 98 100 100 97 0 0 100 0
Quintic 2 100 95 0 100 73 52 56 0 0 100 48 0
Rana 2 96 95 13 5 39 100 54 21 35 0 91 95
Rastrigin 2 97 73 81 1 88 0 0 49 0 0 37 0
Ripple01 2 100 32 0 1 64 0 0 2 0 0 32 0
Ripple25 2 100 93 65 9 91 0 0 7 0 0 85 0
Rosenbrock 2 100 0 0 100 65 0 100 51 0 100 2 0
RosenbrockModified 2 60 4 0 2 10 55 0 10 0 0 3 0
RotatedEllipse01 2 100 98 100 97 77 0 100 1 0 0 100 0
RotatedEllipse02 2 100 100 100 99 83 0 100 9 0 0 100 0
Salomon 2 5 8 95 0 25 0 0 0 0 0 0 0
Sargan 2 100 100 100 100 78 0 100 70 0 0 100 0
Schaffer01 2 3 40 1 2 92 0 0 91 0 0 17 0
Schaffer02 2 15 36 3 2 90 48 0 100 0 100 30 0
Schaffer03 2 23 39 0 7 90 44 0 100 0 0 15 0
Schaffer04 2 13 31 2 1 90 32 0 99 0 0 13 2
SchmidtVetters 3 100 98 100 88 100 100 100 100 21 0 58 98
Schwefel01 2 100 100 100 100 91 0 100 100 0 0 100 0
Schwefel02 2 100 100 100 100 98 0 100 100 0 51 100 0
Schwefel04 2 100 100 100 100 69 0 100 100 0 100 100 0
Schwefel06 2 100 0 0 100 77 0 100 0 0 0 3 0
Schwefel20 2 100 94 0 100 69 0 100 0 0 48 12 0
Schwefel21 2 100 88 0 100 70 0 100 0 0 0 10 0
Schwefel22 2 98 98 0 100 65 0 100 0 0 100 6 0
Schwefel26 2 80 84 10 10 72 52 100 51 53 0 81 99
Schwefel36 2 0 100 100 28 20 0 100 100 0 0 89 0
Shekel05 4 100 0 46 36 18 0 100 48 0 0 52 0
Shekel07 4 100 0 52 38 34 0 100 60 0 0 60 0
Shekel10 4 100 0 52 34 42 0 100 49 0 0 57 0
Shubert01 2 91 100 90 20 54 0 46 57 0 0 16 0
Shubert03 2 82 93 17 5 16 0 100 53 0 0 28 0
Shubert04 2 88 100 89 6 61 100 49 79 0 0 37 0
SineEnvelope 2 9 6 1 0 13 0 0 3 0 0 0 0
SixHumpCamel 2 100 100 100 94 95 100 100 100 0 0 100 0
Sodp 2 100 100 100 100 93 0 100 100 1 0 100 0
Sphere 2 100 100 100 100 92 0 100 100 0 48 100 0
Step 2 0 100 0 88 0 100 0 100 3 0 100 4
Stochastic 2 0 5 0 24 12 0 97 0 0 0 0 0
StretchedV 2 100 0 96 100 76 0 100 100 2 100 31 0
StyblinskiTang 2 100 100 55 31 79 0 100 78 0 0 99 0
TestTubeHolder 2 91 61 42 3 79 100 0 31 0 0 26 0
ThreeHumpCamel 2 100 83 100 82 84 0 100 100 0 0 100 0
Treccani 2 100 100 100 100 91 49 100 100 0 64 100 0
Trefethen 2 1 4 8 1 5 0 0 1 0 0 3 0
Trid 6 100 0 100 100 0 0 100 4 0 0 14 0
Trigonometric01 2 100 100 100 100 42 100 100 100 1 100 86 0
Trigonometric02 2 9 99 59 73 82 0 49 5 0 0 24 0
Tripod 2 99 20 0 44 43 0 100 0 0 50 3 0
Ursem01 2 100 100 60 63 100 100 100 99 0 0 100 0
Ursem03 2 43 100 3 16 71 0 100 0 0 42 4 18
Ursem04 2 100 93 0 50 68 0 100 2 0 100 9 0
UrsemWaves 2 100 6 100 47 1 0 0 61 0 0 31 0
VenterSobiezcczanskiSobieski 2 61 100 62 9 85 0 100 1 0 0 29 0
Vincent 2 100 100 100 100 87 0 100 98 0 0 50 0
Watson 6 100 0 0 43 0 0 0 0 0 0 0 0
Wavy 2 39 80 8 2 91 0 0 63 0 0 63 0
WayburnSeader01 2 100 6 0 100 72 0 38 50 0 98 47 0
WayburnSeader02 2 100 4 0 96 87 0 0 5 0 0 5 0
Weierstrass 2 0 57 0 89 85 0 0 0 0 0 0 0
Whitley 2 5 2 28 2 15 0 0 0 0 0 2 0
Wolfe 3 100 0 100 99 0 0 0 100 0 0 100 0
XinSheYang01 2 0 24 0 13 59 2 41 7 0 0 2 0
XinSheYang02 2 63 35 0 16 51 0 100 0 0 0 3 0
XinSheYang03 2 5 3 0 0 0 0 0 5 0 0 0 0
XinSheYang04 2 40 53 0 4 68 0 0 0 0 0 6 0
Xor 9 100 0 55 77 56 0 0 41 0 0 71 7
YaoLiu04 2 100 100 0 100 85 0 100 36 0 100 31 1
YaoLiu09 2 97 67 79 6 83 0 0 47 0 0 43 0
Zacharov 2 100 100 100 100 82 0 100 100 0 0 100 0
ZeroSum 2 0 0 0 70 0 0 0 0 0 0 6 0
Zettl 2 100 100 100 100 89 0 100 100 0 0 100 0
Zimmerman 2 1 5 0 43 15 0 0 0 0 0 0 0
Zirilli 2 100 100 100 78 97 100 100 100 0 0 99 0

The following table is a split-by-benchmark function of the first table, showing the average number of functions evaluations for successful optimizations only, considering 100 random starting points.

Optimization algorithms performances (N-dimensional)
Function Name N AMPGO ASA BasinHopping CMA-ES CRS2 DE DIRECT Firefly Galileo MLSL PSWARM SIMANN
Ackley 2 1515 1288 1691 468 1203 2019 234 2000 83 2586 349 1186
Adjiman 2 14 1254 639 241 276 1453 176 392 186 2347 451 2001
Alpine01 2 251 1267 2002 618 986 2019 511 1999 95 1064 874 910
Alpine02 2 271 1207 575 291 600 823 218 1156 158 3358 1179 1188
AMGM 2 15 471 466 183 885 360 2001 28 63 2255 783 2001
BartelsConn 2 102 1234 2033 660 811 2019 322 1976 92 1634 344 1496
Beale 2 33 857 1645 570 573 2019 600 1452 89 5142 1427 1597
Bird 2 85 1224 883 351 639 1377 253 1851 122 3130 1081 821
Bohachevsky 2 250 1286 1229 464 748 2019 290 1788 91 2207 335 1063
BoxBetts 3 29 4589 1940 578 612 2018 196 870 111 6707 3518 2001
Branin01 2 11 1265 875 404 558 969 264 1322 92 2990 1276 1221
Branin02 2 290 955 713 378 584 1011 620 1929 96 3974 1125 1099
Brent 2 2 1223 241 477 1190 2019 278 1366 107 2069 396 1360
Brown 2 14 1197 591 354 483 2019 156 1020 97 2123 344 1108
Bukin02 2 8 1242 488 40 380 46 698 23 24 2829 466 2001
Bukin04 2 68 1371 2025 713 701 360 2001 1759 59 59 469 1836
Bukin06 2 842 1226 2047 965 897 381 2001 2000 87 239 1028 877
CarromTable 2 101 1266 495 305 906 213 210 410 86 5512 460 2001
Chichinadze 2 1209 1214 1162 536 641 2019 174 1974 109 3679 496 1222
Cigar 2 26 1245 2018 872 899 2019 258 2000 942 500 1474
Cola 17 1360 60687 2018 2006 2001 2055 2001 2000 927 7979 4021 669
Colville 4 125 7929 2041 244 2003 371 826 196 111 746 1495 1167
Corana 4 2026 576 542 814 2001 1447 2001 1948 243 3749 643 1159
CosineMixture 2 3 1284 208 8 2001 46 292 20 29 2868 415 2001
CrossInTray 2 924 1239 522 308 652 2019 204 1106 146 6148 346 2001
CrossLegTable 2 2007 1255 1615 914 583 2019 138 2000 191 276 2262 793
CrownedCross 2 2020 1254 2969 917 634 2019 190 2000 85 267 2194 2001
Csendes 2 21 516 1046 180 136 2019 40 65 70 2201 299 2001
Cube 2 58 738 2015 1199 938 473 1379 1933 61 977 1368 1698
Damavandi 2 1827 1270 505 379 586 360 2001 2000 61 2267 241 1260
Deb01 2 35 1229 1368 559 1248 360 2001 1535 115 1060 2547 1264
Deb02 2 113 1220 1103 456 751 444 110 1355 175 167 3085 1954
Decanomial 2 522 37 2015 735 564 2019 92 1926 1076 359 251
Deceptive 2 3017 1260 7607 590 617 2019 248 2000 178 775 1084 2001
DeckkersAarts 2 44 1304 1059 303 936 791 490 1269 104 5380 469 1272
DeflectedCorrugatedSpring 2 700 783 361 729 360 2001 1637 60 4221 285 1679
DeVilliersGlasser01 4 231 14114 1963 1985 1908 1274 2001 2000 1783 3951 1934
DeVilliersGlasser02 5 2139 47968 1994 2002 1774 1737 2001 2000 404 2303 3914 1692
DixonPrice 2 17 1330 1106 448 663 1607 431 1482 87 2003 356 1598
Dolan 5 86 43089 2047 1513 2001 1842 884 1992 167 866 3140 1047
DropWave 2 1677 1138 770 451 854 720 174 1766 152 4307 313 1950
Easom 2 1949 1226 1034 420 1207 2019 180 2000 84 3199 411 1019
EggCrate 2 556 1302 744 383 667 2019 270 1617 102 1388 391 909
EggHolder 2 1381 1269 680 494 545 1011 600 1947 169 7069 1123 908
ElAttarVidyasagarDutta 2 109 1314 1306 630 886 1456 728 1998 4579 397 1975
Exp2 2 164 1301 370 327 567 2019 8 1997 90 2735 1627 877
Exponential 2 6 1264 431 304 452 1999 128 776 158 2103 298 2001
FreudensteinRoth 2 89 1126 1554 620 656 2019 719 1937 105 1124 907 1934
Gear 4 1555 414 101 663 555 1725 2001 146 150 2543 1158 2001
Giunta 2 45 1234 693 323 454 2019 219 921 93 2231 318 2001
GoldsteinPrice 2 56 1144 1708 439 668 1537 348 1643 88 4380 331 1690
Griewank 2 1990 1277 569 412 1037 845 348 1997 91 2556 410 1092
Gulf 3 2037 4382 127 729 1360 1522 522 2000 98 6276 3990 281
Hansen 2 859 1228 1064 427 708 1140 252 1396 140 6221 2824 734
Hartmann3 3 45 4299 1638 626 870 2018 230 1847 283 6698 593 2001
Hartmann6 6 69 19805 1996 1426 1962 2016 750 1800 532 7741 1235 2001
HelicalValley 3 33 1936 1057 1979 1772 1511 2000 110 869 3411 835
HimmelBlau 2 15 893 410 686 2019 271 1598 97 2216 1678 1199
HolderTable 2 92 1279 599 300 260 89 224 289 45 6272 448 2001
Holzman 3 194 0 1245 9 2001 66 2001 20 335 4015 251
Hosaki 2 162 1252 779 230 538 739 302 800 158 2465 347 2001
Infinity 2 16 531 1052 183 138 2019 40 64 72 2197 299 2001
JennrichSampson 2 32 1381 2019 531 805 927 524 1906 104 7842 545 914
Judge 2 38 1269 1175 349 731 2019 526 1614 72 4009 606 1424
Katsuura 2 2 1281 177 1271 2001 2019 994 54 104 115 413 2001
Keane 2 6 977 524 8 800 47 2001 20 23 1585 3451 2001
Kowalik 4 814 7965 1984 1721 1347 1966 625 2000 160 7766 3886 1995
Langermann 2 865 1245 1123 409 615 1201 178 1662 124 5780 770 1977
LennardJones 6 82 23343 242 1614 1991 2016 508 904 283 722 2089 2001
Leon 2 34 853 2015 634 574 1707 1311 1687 84 2813 2491 1473
Levy03 2 198 1255 926 383 578 1056 987 1233 88 2281 297 1513
Levy05 2 1717 1268 1306 495 932 1360 218 1745 100 4582 960 825
Levy13 2 430 1242 1554 559 691 1056 1035 1630 81 1831 282 1127
Matyas 2 6 1374 607 411 551 1162 272 1078 61 2193 454 1915
McCormick 2 28 1274 690 352 407 2019 242 1127 140 2504 373 2001
Michalewicz 2 352 1238 1327 292 596 739 144 1196 135 3666 332 942
MieleCantrell 4 180 6000 1801 529 309 2012 269 1259 162 7637 2980 1951
Mishra01 2 2 1200 152 10 2001 1201 674 45 82 2091 3658 2001
Mishra02 2 2 1270 176 18 2001 1789 480 84 107 2164 381 2001
Mishra03 2 1499 1154 2041 930 414 1044 421 2000 95 285 2010 1240
Mishra04 2 1722 1194 2044 899 849 626 357 1982 95 247 1968 1266
Mishra05 2 558 1285 1643 720 372 602 494 1681 98 5781 1079 2001
Mishra06 2 219 1244 899 406 664 1614 250 1527 122 3716 353 1393
Mishra07 2 9 1169 583 400 673 1152 262 824 81 322 1675 1414
Mishra08 2 472 29 2016 732 547 2019 91 1938 1672 369 251
Mishra09 3 177 0 2024 1034 1700 1626 469 1775 288 2811 251
Mishra10 2 2139 58 111 144 2001 381 2001 66 67 2565 652 850
Mishra11 2 13 1168 667 397 616 791 213 611 91 2612 1256 1672
MultiModal 2 58 1042 2028 498 481 2019 160 808 94 677 337 1418
NeedleEye 2 46 649 2020 455 2001 1372 332 1932 100 2634 311 1074
NewFunction01 2 1345 1142 1975 897 885 528 324 1946 93 232 1667 1702
NewFunction02 2 1557 1151 1988 906 476 519 279 1988 81 190 2489 1730
NewFunction03 2 432 1337 1118 464 341 881 349 510 95 4316 386 2001
OddSquare 2 1957 1033 245 146 1036 1649 62 1377 82 4554 406 1979
Parsopoulos 2 19 1260 497 331 515 2019 184 1194 100 2501 2544 1969
Pathological 2 1684 1240 751 527 1878 360 142 2000 61 5014 797 1708
Paviani 10 29 79149 1357 1471 2001 2023 2001 2000 1037 8139 2973 258
Penalty01 2 143 1278 783 497 632 2019 289 1395 78 2623 353 1754
Penalty02 2 1393 1251 1752 690 810 2019 401 1901 77 2365 362 1533
PenHolder 2 166 1264 574 252 307 76 138 128 59 6024 545 291
PermFunction01 2 18 1597 432 616 549 380 1173 85 219 2415 1709
PermFunction02 2 17 1549 440 756 1137 250 1574 98 2954 848 1284
Pinter 2 628 1298 1183 460 716 2019 490 1751 82 1908 336 925
Plateau 2 1669 17 101 152 2001 360 2001 32 65 2880 242 747
Powell 4 36 11842 2014 1071 1909 1069 1014 1989 171 7419 3918 1136
PowerSum 4 998 11953 2123 1927 2001 1520 934 2000 186 7915 3991 1426
Price01 2 6 1272 368 575 902 2019 432 1994 92 1092 382 994
Price02 2 1307 1255 496 334 827 1281 164 1771 98 2707 2297 1728
Price03 2 93 1282 1918 575 877 1172 612 1991 81 2336 1450 1367
Price04 2 48 1117 2009 735 718 1520 428 1691 1831 2350 1864
Qing 2 27 1321 963 621 883 2019 566 1998 2414 390 1316
Quadratic 2 6 1283 651 301 813 1378 429 1758 123 2678 328 1048
Quintic 2 193 1272 2060 709 1182 1156 1304 2000 81 562 291 1550
Rana 2 581 1273 428 452 1427 445 491 1641 116 6758 792 974
Rastrigin 2 529 1268 1278 454 876 1240 278 1902 107 5477 391 857
Ripple01 2 215 1242 2013 668 1002 591 208 1997 151 1759 1100 1650
Ripple25 2 368 1271 1130 482 697 2019 216 1954 163 2187 1139 1928
Rosenbrock 2 53 694 2013 849 876 717 506 1824 86 943 2671 1516
RosenbrockModified 2 1357 725 2000 649 832 971 655 1834 88 7675 2999 1372
RotatedEllipse01 2 10 1324 505 606 891 1514 476 1999 73 2215 390 1035
RotatedEllipse02 2 6 1322 522 573 834 2019 448 1989 73 2035 383 1032
Salomon 2 1955 1163 953 421 1068 1554 152 2000 76 3446 337 971
Sargan 2 6 1311 505 509 725 2019 478 1898 94 2159 363 1051
Schaffer01 2 1977 1089 542 1101 806 1756 88 1192 94 3862 388 339
Schaffer02 2 1908 1248 618 1274 683 1777 152 942 73 215 2063 369
Schaffer03 2 1839 1291 662 1131 1025 940 90 959 95 5682 2161 334
Schaffer04 2 1878 1315 549 1149 1076 1811 96 1004 100 6097 2248 289
SchmidtVetters 3 24 37 1134 11 736 67 186 20 24 9572 1890 286
Schwefel01 2 31 1167 1062 409 550 2019 176 1169 82 2322 371 1256
Schwefel02 2 4 950 352 461 386 2019 234 1072 72 1306 471 1538
Schwefel04 2 14 1318 917 416 471 2019 282 1305 109 231 394 1707
Schwefel06 2 309 1047 2064 769 1230 2019 755 2000 83 4114 474 880
Schwefel20 2 171 1261 2061 720 1108 2019 448 2000 91 3131 371 850
Schwefel21 2 255 1275 2061 711 1118 2019 476 2000 77 1819 372 844
Schwefel22 2 302 1274 2059 1074 1140 2019 562 2000 98 837 380 1009
Schwefel26 2 730 1266 428 443 680 1091 423 1232 74 7057 771 785
Schwefel36 2 2003 1252 673 467 1691 2019 464 58 103 2159 1071 2001
Shekel05 4 253 8054 1048 680 1886 1971 428 1859 319 6822 1064 2000
Shekel07 4 151 8281 1114 624 1897 1971 280 1491 330 7039 947 1991
Shekel10 4 201 8242 1104 655 1893 2012 278 1571 358 6963 1049 1994
Shubert01 2 629 1195 1145 452 889 1217 227 1932 134 5879 3209 725
Shubert03 2 878 1244 703 427 868 1678 223 1928 138 5904 2521 849
Shubert04 2 634 1232 1088 400 796 1351 216 1873 131 6361 2576 846
SineEnvelope 2 1918 1117 451 269 1026 1199 104 1994 92 5153 383 544
SixHumpCamel 2 26 1250 946 261 647 882 232 1380 90 2502 321 1746
Sodp 2 11 1030 1195 307 367 2019 134 594 73 2134 301 2001
Sphere 2 4 1296 353 365 549 2019 224 1201 84 1176 318 1593
Step 2 2032 111 101 285 2001 610 2001 267 86 2027 280 989
Stochastic 2 2013 840 2015 1760 1942 935 820 2000 92 89 491 892
StretchedV 2 9 574 360 823 573 202 870 86 292 2592 2001
StyblinskiTang 2 79 1242 787 384 639 2019 352 1641 149 2515 382 1073
TestTubeHolder 2 927 1286 630 343 946 1356 140 1849 174 7012 1322 945
ThreeHumpCamel 2 34 1247 717 379 564 2019 404 1230 96 2400 322 1916
Treccani 2 12 1262 927 392 603 896 235 1275 94 951 334 1750
Trefethen 2 2009 1249 1696 776 1083 1288 212 1997 82 1833 320 1039
Trid 6 14 14136 1335 1279 2002 2016 1342 1993 592 7767 2421 859
Trigonometric01 2 5 1086 205 356 1291 402 176 442 95 135 808 2001
Trigonometric02 2 1942 1264 1955 688 902 2019 421 1994 83 264 379 991
Tripod 2 496 1257 2050 704 1178 2019 402 2000 100 2055 1583 908
Ursem01 2 29 1258 635 211 535 529 236 759 166 2229 346 2001
Ursem03 2 1603 1240 1996 531 968 2019 336 2000 152 5073 527 1843
Ursem04 2 128 1259 1267 471 829 2019 304 1984 136 543 318 2001
UrsemWaves 2 62 1208 539 393 1225 2019 198 852 173 3375 721 2001
VenterSobiezcczanskiSobieski 2 1325 1277 920 437 950 2019 480 1999 100 4353 375 941
Vincent 2 11 1193 674 370 667 2019 296 1279 153 4013 2135 2001
Watson 6 78 21209 2024 1928 2001 2016 2001 2000 253 7975 4007 1482
Wavy 2 1740 1280 1162 461 788 2019 174 1663 93 6633 455 501
WayburnSeader01 2 40 1292 2019 680 749 1774 566 1848 100 576 1968 2001
WayburnSeader02 2 50 1315 2015 692 916 1653 984 1995 2632 2974 1414
Weierstrass 2 2050 1271 2083 628 1557 2019 250 2000 106 267 299 1082
Whitley 2 1958 1269 1814 684 1031 843 496 2000 79 297 462 1300
Wolfe 3 6 4586 449 932 2001 2018 676 107 146 2837 483 2001
XinSheYang01 2 2012 931 2014 1875 1548 1146 1616 1965 97 88 319 1146
XinSheYang02 2 1299 1208 2041 430 903 2019 418 2000 96 4950 344 1556
XinSheYang03 2 1909 130 142 134 34 2019 16 1980 88 3034 625 2001
XinSheYang04 2 1609 1249 566 344 1097 2019 170 2000 87 4148 775 1984
Xor 9 51 48195 1700 996 1391 1823 1104 1363 221 8438 1388 1658
YaoLiu04 2 60 1202 2033 593 575 1109 349 1913 84 299 798 1632
YaoLiu09 2 545 1252 1268 442 888 1227 278 1910 112 2606 398 908
Zacharov 2 11 1314 716 402 576 2019 290 1335 113 2354 337 1721
ZeroSum 2 2169 1212 2042 966 740 704 2001 2000 80 282 790 879
Zettl 2 22 1272 1511 430 489 2019 212 1085 92 3940 395 1991
Zimmerman 2 1987 1244 2047 1183 1369 2019 672 2000 106 927 449 1152
Zirilli 2 35 1238 763 241 594 660 257 815 74 2325 334 1668

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