# SciPy Extended¶

Fopt Known Xopt Known Difficulty
Yes Yes Easy

All the test functions in this benchmark suite have been taken from the mathematical literature on Global Optimization. The test suite currently contains:

1. 18 one-dimensional test functions with multiple local/global minima.
2. 235 multivariate problems (where the number of independent variables ranges from 2 to 17), again with multiple local/global minima. I have added about 40 new functions to the standard SciPy benchmarks.

For the purpose of this exercise, the 1D functions have been excluded from the benchmarks.

The index of the test function is in SciPy Test Functions Index page: as the list is quite large, their definition has been split into multiple pages using the first letter of their name. Whenever possible, a 3D plot of the test function has been provided.

As an example, Figure 1.1 below contains some 3D representations of some of the benchmark functions in the SciPy Extended test suite.

 Alpine 1 Corana Deceptive Gramacy Lee 3 Langermann OddSquare

Note

If you wish to contribute to the test suite (i.e., to add a new benchmark problem), please do send me an email to andrea.gavana@gmail.com, I’ll integrate your contribution with due credits and I will re-run the algorithms comparison.

## Methodology¶

Most of the classical test functions found in the literature suffer from a number of limitations and weaknesses, that are often exploited by global optimization algorithms:

1. Initialization Bias (Central Bias): many of the benchmark functions in the SciPy test suite have bounds that are symmetric with respect to the global optimum (i.e., the global optimum is exactly in the middle), or they have one or more optima on the bounds.
2. Axial and Directional Bias: many mathematical functions used for benchmarking exhibit some alignment in the structure, and in particular valleys containing local minima.
3. Rotational Invariance: some mathematical functions, such as Schaffer’s F6 function, exhibit rotational symmetry.
4. Regularity: many elementary benchmark functions have local minima spread in regular patterns.

For all these reasons, I now actually prefer test function generators to the standard, classical benchmarks, although the latter are so much more beautiful to look at :-) .

That said, one approach I have taken to break some of the issues above is to shift the global optimum of all functions to a new, random point inside the function domain. One possibility for doing that is kind-of explained in the paper Novel composition test functions for numerical global optimization, and it can be sketched as follow:

1. Start with the original function , and leave the original bounds untouched.

2. If we define to be the original global optimum and the new, randomly generated global optimum, then we can define a new function as:

which we will then use as our optimization target function. Of course, by doing that we risk that some of the points passed by the optimizer to the new objective function will be translated outside the original domain: to avoid this, I have introduced a penalty function for when the evaluation point goes out of the specified bounds.

If we define:

Then our penalty function becomes:

## General Solvers Performances¶

Table 1.1 below shows the overall success of all Global Optimization algorithms, considering for every benchmark function 100 random starting points, for a maximum allowable budget of .

As said in the introduction, quite a few solvers do not really support an initial starting point, but life is tough :-) .

Looking at the table, for example, BiteOpt comes out as the winner solving 83.4% of the problems using, on average, around 430 functions evaluations. MCS comes close second at 81.9% solved benchmarks but with a much lower budget at 235 functions evaluations.

Note

The reported number of functions evaluations refers to successful optimizations only.

Table 1.1: Solvers performances on the SciPy Extended benchmark suite at NF = 2,000
Optimization Method Overall Success (%) Functions Evaluations
AMPGO 69.47% 321
BasinHopping 64.68% 314
BiteOpt 83.40% 430
CMA-ES 58.78% 570
CRS2 73.41% 806
DE 75.39% 1,059
DIRECT 72.50% 376
DualAnnealing 72.42% 256
LeapFrog 61.85% 314
MCS 81.95% 235
PSWARM 21.19% 1,420
SCE 71.16% 522
SHGO 69.75% 252

These results are also depicted in Figure 1.2, which shows that BiteOpt is the better-performing optimization algorithm, closely followed by MCS. Excluding PSWARM, all the solvers were able to reach the global optimum for at least 50% of the problems, on average.

Figure 1.2: Optimization algorithms performances on the SciPy Extended test suite at

Pushing the available budget to a very generous , the results do not change that much in terms of ranking between the two best solvers (BiteOpt and MCS), but there is a significant uptick in performances for PSWARM and AMPGO, while a less pronounced one can be seen for SCE, as shown in Table 1.2 and Figure 1.3.

Table 1.2: Solvers performances on the SciPy Extended benchmark suite at NF = 10,000
Optimization Method Overall Success (%) Functions Evaluations
AMPGO 80.67% 1,023
BasinHopping 69.32% 553
BiteOpt 89.85% 728
CMA-ES 61.76% 696
CRS2 79.76% 1,035
DE 84.25% 1,442
DIRECT 77.88% 666
DualAnnealing 77.16% 441
LeapFrog 62.70% 361
MCS 88.68% 578
PSWARM 75.93% 2,626
SCE 79.97% 958
SHGO 77.05% 692

These results are also depicted in Figure 1.3, which shows the dramatic improvement of the PSWARM algorithm when given enough budget of functions evaluations - and this is also reflected on the average number of functions evaluations for solved problems (2,626). Similar, but less dramatic conclusions can be reached for AMPGO.

Figure 1.3: Optimization algorithms performances on the SciPy Extended test suite at

## Sensitivities on Functions Evaluations Budget¶

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 100, 200, 300,... 2000, 5000, 10000.

In order to do that, we can present the results using two different types of visualizations. The first one is some sort of “small multiples” in which each solver gets an individual subplot showing the improvement in the number of solved problems as a function of the available number of function evaluations - on top of a background set of grey, semi-transparent lines showing all the other solvers performances.

This visual gives an indication of how good/bad is a solver compared to all the others as function of the budget available. Results are shown in Figure 1.4.

Figure 1.4: Percentage of problems solved given a fixed number of function evaluations on the SciPy Extended test suite

The second type of visualization is sometimes referred as “Slopegraph” and there are many variants on the plot layout and appearance that we can implement. The version shown in Figure 1.5 aggregates all the solvers together, so it is easier to spot when a solver overtakes another or the overall performance of an algorithm while the available budget of function evaluations changes.

Figure 1.5: Percentage of problems solved given a fixed number of function evaluations on the SciPy Extended test suite

A few obvious conclusions we can draw from these pictures are:

1. For this specific benchmark test suite, if you have a very limited budget in terms of function evaluations, then MCS, SHGO, DualAnnealing and AMPGO are a very good choice: they solve between 35% and 45% of all problems in 100 function evaluations or less, and the percentages go up to 70% if you allow at least 500 functions evaluations.
2. The new entry BiteOpt is not so strong when the budget is very limited, but it quickly overtake almost all the other solvers at the budget mark. In the end it surpasses also MCS at around the cutoff to claim the top spot to the end.
3. The improvement of PSWARM when we allow a more generous budget is phenomenal, going from 20% to 70% solved benchmarks. A similar - but less squished - curve can be seen for DE, which given enough function evaluations surpasses all the other SciPy solvers.

## Tolerance Sensitivities¶

I admit that choosing a tolerance of is kind of arbitrary per se, so I have decided to repeat part of the SciPy Extended test suite with different tolerances, and specifically:

• I have only run the benchmarks at a fixed budget of function evaluations, namely .
• Five different tolerances were chosen, , , , and - the next to last I already had it.

I am not sure I was expecting any surprise here, but in any case Table 1.3 gives a summary of the solvers performances with a variable tolerance.

Table 1.3: Solvers performances on the SciPy Extended benchmark suite with variable tolerances
Tolerance AMPGO BasinHopping BiteOpt CMA-ES CRS2 DE DIRECT DualAnnealing LeapFrog MCS PSWARM SCE SHGO Best Solver
80.2% 69.1% 92.5% 81.6% 86.8% 90.5% 86.6% 85.6% 73.6% 92.9% 85.3% 88.7% 72.2% MCS
71.4% 67.8% 86.2% 71.8% 77.7% 82.2% 76.7% 74.8% 64.2% 85.2% 62.8% 76.9% 71.0% BiteOpt
69.6% 65.1% 84.0% 69.5% 75.1% 77.2% 73.7% 72.4% 62.4% 83.2% 37.8% 73.3% 70.1% BiteOpt
69.5% 64.7% 83.4% 58.8% 73.4% 75.4% 72.5% 72.3% 61.8% 81.9% 21.2% 71.2% 69.7% BiteOpt
67.3% 54.4% 79.9% 54.8% 68.2% 70.5% 67.9% 64.9% 59.5% 80.2% 6.1% 70.0% 63.6% MCS

What we can see from the table is that the two top solvers in this specific benchmark suite keep their crown and stay well above all the other optimization algorithms. In general it is to be expected that by tightening the tolerance less problems will be solved, however some of the algorithms show a much steeper decrease in performances compared to others.

For example, BiteOpt and MCS lose about 12% of the benchmarks when the tolerance goes from to , while DE, DIRECT and DualAnnealing are closer to 20% and PSWARM has a dramatic drop of 80%.

The results from Table 1.3 are visualized in Figure 1.6 below.

Figure 1.6: Percentage of problems solved as a function of tolerance on the SciPy Extended test suite

From the picture above it is even more striking to see the drop of PSWARM.

## Tough Functions and Boring Tables¶

As I did in the previous benchmark, I will dedicate a section to the “toughest” functions in this benchmark suite. The test suite contains a variety of Global Optimization problems, some of them are harder to solve than others, irrespectively of the algorithm chosen to minimize the test function.

Table 1.4 has been obtained by running all the Global Optimizers available against all the N-D test functions for a collection of 100 random starting points with a maximum budget of , and then averaging the successful minimizations across all the optimizers.

Table 1.4: SciPy Extended benchmark functions “hardness”
Test Function N Overall Success (%)
HappyCat 2 0.15%
Bukin06 2 0.31%
Cola 17 1.00%
CrownedCross 2 7.31%
CrossLegTable 2 7.54%
Meyer 3 8.23%
Paviani 10 9.77%
Thurber 7 9.85%
SineEnvelope 2 13.31%
Peaks 2 14.69%
Trefethen 2 14.77%
Whitley 2 14.92%
NewFunction02 2 15.38%
Mishra04 2 15.38%
Mishra03 2 15.38%
DeVilliersGlasser02 5 15.38%
BiggsExp06 6 15.62%
Zagros 2 18.00%
Xor 9 23.08%
EggHolder 2 23.08%
BiggsExp05 5 26.15%
Zimmerman 2 28.62%
XinSheYang03 2 30.15%
NewFunction01 2 30.77%
Rana 2 30.77%
Osborne 5 30.77%
Salomon 2 31.69%
PowerSum 4 31.92%
Griewank 2 33.62%
Damavandi 2 33.62%
Ripple01 2 34.38%
Stochastic 2 36.38%
Kowalik 4 41.46%
Hougen 5 42.23%
Schaffer01 2 43.38%
Schaffer04 2 46.00%
XinSheYang01 2 47.69%
Ackley04 2 48.85%
Schaffer03 2 51.54%
Schaffer02 2 51.69%
Watson 6 53.54%
Deceptive 2 53.69%
UrsemWaves 2 53.85%
DropWave 2 54.23%
ZeroSum 2 54.85%
Spike 2 55.23%
OddSquare 2 55.54%
RosenbrockModified 2 57.69%
MeyerRoth 3 58.85%
Corana 4 59.15%
Colville 4 59.31%
Weierstrass 2 60.69%
Ratkowsky01 4 60.92%
Easom 2 61.00%
DeVilliersGlasser01 4 61.54%
Simpleton 10 61.54%
XinSheYang04 2 63.46%
Gulf 3 63.69%
Price02 2 64.69%
Shekel10 4 65.08%
AMGM 2 65.54%
InvertedCosine 2 65.69%
TestTubeHolder 2 66.15%
Eckerle4 3 67.08%
DeJong5 2 67.46%
MultiGaussian 2 67.85%
DeflectedCorrugatedSpring 2 68.77%
Shekel07 4 69.08%
Shekel05 4 69.92%
Ackley01 2 70.00%
Price03 2 70.77%
Levy05 2 71.92%
Penalty02 2 72.23%
Crescent 2 72.62%
Tripod 2 73.31%
BiggsExp04 4 73.85%
GramacyLee03 2 74.38%
Shubert03 2 75.38%
Wavy 2 75.46%
Ursem03 2 75.69%
Dolan 5 75.77%
XinSheYang02 2 76.00%
Pathological 2 76.15%
Rastrigin 2 76.69%
Alpine02 2 76.85%
SchmidtVetters 3 76.92%
Mishra05 2 76.92%
CarromTable 2 76.92%
Hansen 2 78.00%
Branin02 2 79.08%
BuecheRastrigin 2 79.08%
Ripple25 2 79.08%
Mishra06 2 79.08%
YaoLiu09 2 79.54%
Schwefel26 2 79.54%
Trid 6 80.08%
LunacekBiRastrigin 2 80.77%
Powell 4 81.46%
Tsoulos 2 82.69%
Mishra10 2 82.77%
VenterSobiezcczanskiSobieski 2 82.85%
Trigonometric02 2 82.92%
Sinusoidal 2 82.92%
SphericalSinc 2 83.00%
Gear 4 83.00%
Shubert01 2 83.69%
Deb03 2 83.69%
Shubert04 2 84.23%
F2 2 84.46%
HolderTable01 2 84.46%
HolderTable02 2 84.62%
Schwefel36 2 84.62%
PenHolder 2 84.62%
GramacyLee02 2 84.62%
FreudensteinRoth 2 85.85%
CrossInTray 2 86.46%
Plateau 2 86.77%
Step01 2 86.85%
Quintic 2 87.08%
ElAttarVidyasagarDutta 2 87.08%
Step02 2 87.23%
Langermann 2 87.31%
Schwefel06 2 87.69%
Pinter 2 88.00%
Vincent 2 88.23%
EggCrate 2 88.46%
Step03 2 89.00%
Bird 2 90.00%
Penalty01 2 90.69%
Mishra09 3 90.85%
Deb01 2 90.92%
StyblinskiTang 2 91.15%
Trigonometric01 2 91.38%
Schwefel21 2 91.85%
Hartmann6 6 92.00%
Schwefel22 2 92.15%
Schwefel20 2 92.23%
MieleCantrell 4 92.31%
RosenbrockDisc 2 92.31%
LennardJones 6 92.31%
Friedman 5 92.31%
Ackley02 2 92.54%
YaoLiu04 2 92.62%
Alpine01 2 92.85%
Levy13 2 93.08%
Ursem04 2 93.08%
Ackley03 2 93.15%
SawtoothXY 2 93.31%
Ratkowsky02 3 93.31%
DeckkersAarts 2 93.54%
HelicalValley 3 93.62%
Engvall 2 93.77%
BoxBetts 3 94.15%
Bohachevsky02 2 94.23%
Michalewicz 2 94.23%
HyperGrid 2 94.69%
Bohachevsky01 2 95.00%
Decanomial 2 95.38%
BartelsConn 2 95.46%
MullerBrown 2 95.54%
Mishra08 2 95.54%
ThreeHumpCamel 2 95.54%
HimmelBlau 2 95.69%
Parsopoulos 2 95.69%
GoldsteinPrice 2 96.15%
Ursem01 2 96.23%
SixHumpCamel 2 96.38%
Levy03 2 96.46%
Cigar 2 96.54%
Branin01 2 96.62%
Beale 2 96.69%
Judge 2 96.77%
Price04 2 96.85%
Zirilli 2 96.92%
StretchedV 2 97.00%
Picheny 2 97.08%
Bohachevsky03 2 97.15%
Hartmann3 3 97.23%
PermFunction01 2 97.54%
BiggsExp03 3 97.62%
Leon 2 97.62%
Price01 2 97.77%
McCormick 2 97.85%
Cube 2 97.85%
Kearfott 2 98.00%
Qing 2 98.08%
Bukin04 2 98.38%
Giunta 2 98.46%
Rosenbrock 2 98.46%
BiggsExp02 2 98.46%
Exp2 2 98.54%
ReduxSum 2 98.62%
Mishra11 2 98.62%
Mishra10b 2 98.69%
Brent 2 98.92%
Treccani 2 98.92%
PermFunction02 2 99.00%
JennrichSampson 2 99.08%
RotatedEllipse01 2 99.08%
Schwefel04 2 99.15%
RotatedEllipse02 2 99.23%
NeedleEye 2 99.23%
DixonPrice 2 99.31%
Mishra07 2 99.46%
LunacekBiSphere 2 99.46%
Zettl 2 99.54%
TridiagonalMatrix 2 99.54%
Exponential 2 99.62%
Keane 2 99.77%
Schwefel01 2 99.77%
Sargan 2 99.77%
Matyas 2 99.77%
Brown 2 99.85%
Zacharov 2 99.85%
Sodp 2 99.85%
Sphere 2 99.85%
Schwefel02 2 99.92%
MultiModal 2 99.92%
Csendes 2 99.92%
Bukin02 2 100.00%
Hosaki 2 100.00%
Booth 2 100.00%
CosineMixture 2 100.00%
Mishra02 2 100.00%
Mishra01 2 100.00%
Katsuura 2 100.00%
Wolfe 3 100.00%
Infinity 2 100.00%

It can be easily seen that the HappyCat, Bukin06 and Cola were always hard problems for all the algorithms, while others like Bukin02 and CosineMixture are amongst the easiest ones.

Table 1.5 is a split-by-benchmark function of the first table, showing the percentage of successful optimizations per benchmark with a maximum budget of

Table 1.5: SciPy Extended benchmark - split by function success
Test Function N AMPGO BasinHopping BiteOpt CMA-ES CRS2 DE DIRECT DualAnnealing LeapFrog MCS PSWARM SCE SHGO
Ackley01 2 90.0 98.0 100.0 15.0 84.0 95.0 38.0 99.0 14.0 87.0 88.0 96.0 6.0
Ackley02 2 100.0 100.0 100.0 100.0 95.0 100.0 100.0 51.0 100.0 100.0 95.0 100.0 62.0
Ackley03 2 100.0 97.0 100.0 39.0 98.0 100.0 100.0 98.0 82.0 100.0 98.0 99.0 100.0
Ackley04 2 100.0 23.0 43.0 31.0 45.0 23.0 95.0 50.0 15.0 79.0 17.0 14.0 100.0
Adjiman 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Alpine01 2 100.0 92.0 100.0 100.0 98.0 100.0 100.0 32.0 100.0 100.0 100.0 85.0 100.0
Alpine02 2 100.0 37.0 96.0 48.0 40.0 45.0 84.0 81.0 71.0 100.0 97.0 100.0 100.0
AMGM 2 100.0 54.0 100.0 44.0 14.0 1.0 45.0 53.0 42.0 100.0 100.0 99.0 100.0
BartelsConn 2 99.0 100.0 100.0 88.0 100.0 100.0 85.0 93.0 100.0 79.0 97.0 100.0 100.0
Beale 2 100.0 100.0 98.0 83.0 97.0 100.0 100.0 98.0 84.0 100.0 97.0 100.0 100.0
BiggsExp02 2 100.0 99.0 100.0 99.0 96.0 100.0 100.0 100.0 99.0 88.0 99.0 100.0 100.0
BiggsExp03 3 100.0 100.0 100.0 100.0 99.0 100.0 96.0 100.0 91.0 98.0 85.0 100.0 100.0
BiggsExp04 4 61.0 95.0 100.0 100.0 98.0 100.0 3.0 91.0 62.0 94.0 2.0 100.0 54.0
BiggsExp05 5 28.0 24.0 60.0 55.0 40.0 35.0 0.0 4.0 2.0 60.0 0.0 28.0 4.0
BiggsExp06 6 14.0 12.0 56.0 30.0 43.0 0.0 0.0 3.0 0.0 40.0 0.0 4.0 1.0
Bird 2 100.0 56.0 98.0 69.0 88.0 92.0 100.0 99.0 74.0 99.0 96.0 99.0 100.0
Bohachevsky01 2 100.0 100.0 100.0 82.0 98.0 100.0 100.0 100.0 85.0 86.0 100.0 100.0 84.0
Bohachevsky02 2 100.0 100.0 98.0 92.0 98.0 100.0 100.0 98.0 65.0 84.0 98.0 100.0 92.0
Bohachevsky03 2 100.0 100.0 100.0 91.0 99.0 100.0 100.0 100.0 90.0 90.0 99.0 100.0 94.0
Booth 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
BoxBetts 3 100.0 100.0 100.0 100.0 100.0 100.0 99.0 84.0 91.0 100.0 94.0 100.0 56.0
Brad 3 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Branin01 2 100.0 88.0 97.0 91.0 91.0 99.0 100.0 99.0 93.0 100.0 98.0 100.0 100.0
Branin02 2 99.0 58.0 83.0 51.0 64.0 75.0 100.0 94.0 35.0 99.0 79.0 91.0 100.0
Brent 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.0 100.0 92.0 95.0 100.0 100.0
Brown 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 98.0 100.0 100.0
BuecheRastrigin 2 52.0 81.0 100.0 5.0 84.0 92.0 100.0 100.0 24.0 99.0 91.0 100.0 100.0
Bukin02 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Bukin04 2 99.0 100.0 100.0 98.0 100.0 100.0 100.0 87.0 100.0 97.0 98.0 100.0 100.0
Bukin06 2 4.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
CarromTable 2 100.0 0.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 0.0 100.0
Chichinadze 2 96.0 22.0 100.0 18.0 72.0 99.0 84.0 99.0 31.0 100.0 56.0 81.0 67.0
Cigar 2 95.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 88.0 100.0 72.0
Cola 17 1.0 1.0 0.0 10.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0
Colville 4 28.0 70.0 97.0 51.0 67.0 80.0 1.0 97.0 48.0 99.0 1.0 58.0 74.0
Corana 4 54.0 63.0 100.0 11.0 85.0 100.0 21.0 100.0 2.0 94.0 53.0 85.0 1.0
CosineMixture 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Crescent 2 100.0 20.0 100.0 98.0 91.0 100.0 17.0 6.0 100.0 63.0 92.0 100.0 57.0
CrossInTray 2 99.0 19.0 100.0 36.0 93.0 100.0 100.0 100.0 88.0 100.0 96.0 93.0 100.0
CrossLegTable 2 2.0 0.0 13.0 6.0 13.0 5.0 20.0 0.0 30.0 9.0 0.0 0.0 0.0
CrownedCross 2 6.0 0.0 19.0 5.0 11.0 7.0 7.0 0.0 17.0 22.0 1.0 0.0 0.0
Csendes 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.0 100.0 100.0
Cube 2 100.0 100.0 100.0 100.0 100.0 100.0 91.0 100.0 100.0 100.0 81.0 100.0 100.0
Damavandi 2 31.0 3.0 34.0 3.0 9.0 18.0 55.0 31.0 6.0 98.0 16.0 33.0 100.0
Deb01 2 100.0 100.0 100.0 96.0 84.0 100.0 100.0 100.0 96.0 100.0 93.0 13.0 100.0
Deb03 2 100.0 81.0 100.0 75.0 52.0 59.0 87.0 92.0 72.0 100.0 93.0 77.0 100.0
Decanomial 2 98.0 100.0 100.0 91.0 92.0 100.0 98.0 100.0 100.0 100.0 61.0 100.0 100.0
Deceptive 2 77.0 27.0 100.0 21.0 38.0 78.0 77.0 18.0 27.0 93.0 76.0 58.0 8.0
DeckkersAarts 2 99.0 79.0 100.0 93.0 89.0 100.0 100.0 78.0 85.0 95.0 98.0 100.0 100.0
DeflectedCorrugatedSpring 2 59.0 100.0 69.0 1.0 21.0 59.0 100.0 83.0 5.0 100.0 97.0 100.0 100.0
DeJong5 2 81.0 7.0 100.0 14.0 66.0 83.0 100.0 100.0 12.0 95.0 41.0 98.0 80.0
DeVilliersGlasser01 4 100.0 100.0 100.0 100.0 0.0 100.0 0.0 100.0 0.0 100.0 0.0 0.0 100.0
DeVilliersGlasser02 5 100.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0 0.0
DixonPrice 2 100.0 100.0 100.0 100.0 95.0 100.0 100.0 100.0 97.0 100.0 99.0 100.0 100.0
Dolan 5 100.0 100.0 100.0 100.0 2.0 0.0 100.0 99.0 100.0 100.0 84.0 0.0 100.0
DropWave 2 17.0 100.0 57.0 3.0 35.0 55.0 24.0 56.0 10.0 97.0 93.0 71.0 87.0
Easom 2 60.0 0.0 100.0 1.0 95.0 100.0 17.0 60.0 50.0 100.0 15.0 95.0 100.0
Eckerle4 3 100.0 25.0 100.0 11.0 94.0 99.0 2.0 99.0 41.0 92.0 55.0 96.0 58.0
EggCrate 2 100.0 59.0 100.0 34.0 98.0 99.0 100.0 100.0 63.0 100.0 97.0 100.0 100.0
EggHolder 2 100.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0 100.0
ElAttarVidyasagarDutta 2 100.0 58.0 100.0 51.0 87.0 91.0 100.0 100.0 63.0 96.0 97.0 100.0 89.0
Engvall 2 100.0 100.0 100.0 19.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Exp2 2 100.0 99.0 100.0 100.0 91.0 100.0 100.0 100.0 99.0 92.0 100.0 100.0 100.0
Exponential 2 100.0 100.0 100.0 100.0 98.0 100.0 100.0 100.0 100.0 100.0 97.0 100.0 100.0
F2 2 94.0 90.0 100.0 31.0 79.0 95.0 100.0 100.0 24.0 99.0 86.0 100.0 100.0
FreudensteinRoth 2 99.0 52.0 99.0 55.0 87.0 89.0 100.0 100.0 52.0 87.0 98.0 98.0 100.0
Friedman 5 100.0 100.0 100.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Gear 4 100.0 3.0 100.0 99.0 99.0 100.0 100.0 100.0 61.0 100.0 100.0 100.0 17.0
Giunta 2 100.0 100.0 100.0 88.0 96.0 100.0 100.0 100.0 100.0 100.0 96.0 100.0 100.0
GoldsteinPrice 2 100.0 99.0 96.0 82.0 90.0 100.0 100.0 100.0 86.0 100.0 97.0 100.0 100.0
GramacyLee02 2 100.0 0.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
GramacyLee03 2 100.0 49.0 100.0 33.0 56.0 54.0 100.0 100.0 27.0 96.0 68.0 84.0 100.0
Griewank 2 53.0 0.0 50.0 2.0 62.0 47.0 17.0 46.0 5.0 93.0 38.0 3.0 21.0
Gulf 3 95.0 40.0 100.0 70.0 90.0 100.0 0.0 43.0 82.0 93.0 0.0 100.0 15.0
Hansen 2 82.0 94.0 91.0 20.0 67.0 95.0 75.0 92.0 31.0 100.0 89.0 78.0 100.0
HappyCat 2 0.0 0.0 0.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0
Hartmann3 3 100.0 100.0 98.0 98.0 97.0 100.0 100.0 97.0 81.0 100.0 93.0 100.0 100.0
Hartmann6 6 100.0 100.0 93.0 94.0 96.0 97.0 94.0 93.0 75.0 95.0 67.0 99.0 93.0
HelicalValley 3 83.0 100.0 100.0 100.0 100.0 100.0 94.0 100.0 96.0 98.0 46.0 100.0 100.0
HimmelBlau 2 100.0 74.0 99.0 91.0 91.0 100.0 100.0 100.0 91.0 100.0 99.0 99.0 100.0
HolderTable01 2 100.0 37.0 100.0 36.0 87.0 98.0 97.0 100.0 47.0 100.0 96.0 100.0 100.0
HolderTable02 2 100.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 0.0 100.0
Hosaki 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Hougen 5 9.0 69.0 98.0 71.0 95.0 22.0 0.0 16.0 0.0 68.0 0.0 64.0 37.0
HyperGrid 2 100.0 100.0 100.0 89.0 90.0 100.0 100.0 100.0 93.0 100.0 96.0 63.0 100.0
Infinity 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
InvertedCosine 2 35.0 42.0 70.0 3.0 50.0 78.0 73.0 98.0 10.0 95.0 100.0 100.0 100.0
JennrichSampson 2 99.0 97.0 100.0 100.0 95.0 100.0 100.0 98.0 99.0 100.0 100.0 100.0 100.0
Judge 2 100.0 98.0 95.0 87.0 96.0 99.0 100.0 98.0 86.0 100.0 99.0 100.0 100.0
Katsuura 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Keane 2 100.0 100.0 100.0 99.0 99.0 100.0 100.0 100.0 100.0 100.0 99.0 100.0 100.0
Kearfott 2 100.0 94.0 100.0 99.0 93.0 100.0 100.0 100.0 91.0 99.0 99.0 99.0 100.0
Kowalik 4 42.0 35.0 62.0 41.0 77.0 79.0 0.0 12.0 33.0 72.0 1.0 73.0 12.0
Langermann 2 85.0 92.0 98.0 32.0 86.0 98.0 100.0 100.0 47.0 100.0 97.0 100.0 100.0
LennardJones 6 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 0.0
Leon 2 100.0 99.0 100.0 96.0 100.0 99.0 100.0 90.0 95.0 99.0 91.0 100.0 100.0
Levy03 2 100.0 77.0 100.0 87.0 99.0 100.0 100.0 100.0 92.0 100.0 99.0 100.0 100.0
Levy05 2 77.0 76.0 90.0 23.0 51.0 80.0 77.0 79.0 19.0 100.0 69.0 95.0 99.0
Levy13 2 99.0 100.0 100.0 61.0 91.0 100.0 100.0 99.0 74.0 99.0 97.0 100.0 90.0
LunacekBiRastrigin 2 69.0 85.0 100.0 9.0 81.0 90.0 100.0 100.0 18.0 99.0 99.0 100.0 100.0
LunacekBiSphere 2 100.0 96.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 97.0 100.0 100.0
Matyas 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 98.0 100.0 99.0
McCormick 2 100.0 82.0 100.0 95.0 98.0 100.0 100.0 100.0 99.0 100.0 98.0 100.0 100.0
Meyer 3 0.0 0.0 37.0 0.0 12.0 12.0 0.0 0.0 0.0 0.0 0.0 46.0 0.0
MeyerRoth 3 72.0 47.0 79.0 56.0 84.0 90.0 3.0 36.0 63.0 97.0 43.0 94.0 1.0
Michalewicz 2 99.0 100.0 100.0 63.0 88.0 100.0 100.0 100.0 77.0 100.0 99.0 99.0 100.0
MieleCantrell 4 99.0 100.0 100.0 96.0 100.0 100.0 99.0 100.0 97.0 100.0 83.0 100.0 26.0
Mishra01 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Mishra02 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Mishra03 2 100.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0 0.0
Mishra04 2 100.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0 0.0
Mishra05 2 100.0 100.0 100.0 0.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 0.0
Mishra06 2 99.0 37.0 93.0 55.0 79.0 96.0 100.0 99.0 54.0 100.0 99.0 17.0 100.0
Mishra07 2 100.0 98.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.0 96.0 100.0
Mishra08 2 96.0 100.0 100.0 92.0 92.0 100.0 99.0 100.0 100.0 99.0 64.0 100.0 100.0
Mishra09 3 100.0 97.0 98.0 92.0 96.0 99.0 100.0 49.0 89.0 100.0 100.0 77.0 84.0
Mishra10 2 100.0 40.0 100.0 65.0 95.0 100.0 100.0 100.0 57.0 100.0 36.0 96.0 87.0
Mishra10b 2 100.0 99.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 98.0 86.0 100.0
Mishra11 2 100.0 96.0 100.0 100.0 99.0 100.0 100.0 100.0 98.0 100.0 100.0 89.0 100.0
MullerBrown 2 100.0 97.0 100.0 74.0 93.0 96.0 100.0 97.0 88.0 98.0 100.0 99.0 100.0
MultiGaussian 2 63.0 92.0 81.0 1.0 33.0 52.0 100.0 81.0 11.0 91.0 86.0 92.0 99.0
MultiModal 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.0 100.0 100.0 100.0
NeedleEye 2 100.0 100.0 100.0 100.0 96.0 100.0 98.0 100.0 100.0 100.0 96.0 100.0 100.0
NewFunction01 2 100.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0 0.0 100.0 0.0 100.0 0.0
NewFunction02 2 100.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0 0.0
OddSquare 2 40.0 3.0 95.0 22.0 79.0 99.0 45.0 45.0 11.0 88.0 93.0 39.0 63.0
Osborne 5 100.0 0.0 100.0 0.0 100.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0 0.0
Parsopoulos 2 100.0 77.0 100.0 95.0 91.0 100.0 100.0 100.0 98.0 100.0 94.0 89.0 100.0
Pathological 2 100.0 41.0 100.0 64.0 92.0 88.0 37.0 97.0 93.0 100.0 88.0 2.0 88.0
Paviani 10 1.0 0.0 100.0 0.0 0.0 0.0 0.0 1.0 0.0 20.0 0.0 5.0 0.0
Peaks 2 92.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 99.0 0.0 0.0 0.0
Penalty01 2 100.0 22.0 100.0 89.0 92.0 100.0 100.0 100.0 78.0 100.0 99.0 100.0 99.0
Penalty02 2 88.0 100.0 98.0 32.0 95.0 93.0 39.0 94.0 49.0 66.0 96.0 77.0 12.0
PenHolder 2 100.0 0.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
PermFunction01 2 100.0 100.0 100.0 98.0 93.0 99.0 100.0 99.0 90.0 100.0 89.0 100.0 100.0
PermFunction02 2 100.0 100.0 99.0 99.0 95.0 100.0 100.0 100.0 97.0 100.0 97.0 100.0 100.0
Picheny 2 100.0 100.0 99.0 95.0 91.0 100.0 100.0 100.0 81.0 99.0 97.0 100.0 100.0
Pinter 2 100.0 86.0 89.0 44.0 83.0 98.0 100.0 94.0 53.0 100.0 97.0 100.0 100.0
Plateau 2 100.0 82.0 100.0 91.0 100.0 100.0 100.0 100.0 92.0 100.0 5.0 100.0 58.0
Powell 4 99.0 100.0 100.0 100.0 100.0 100.0 14.0 53.0 97.0 97.0 12.0 100.0 87.0
PowerSum 4 38.0 94.0 56.0 20.0 21.0 15.0 0.0 13.0 14.0 78.0 0.0 24.0 42.0
Price01 2 99.0 99.0 100.0 88.0 99.0 100.0 100.0 100.0 100.0 98.0 99.0 89.0 100.0
Price02 2 73.0 8.0 71.0 7.0 58.0 62.0 97.0 96.0 14.0 100.0 62.0 93.0 100.0
Price03 2 100.0 76.0 57.0 69.0 48.0 57.0 100.0 70.0 44.0 100.0 44.0 55.0 100.0
Price04 2 99.0 99.0 97.0 96.0 97.0 97.0 98.0 95.0 96.0 100.0 94.0 97.0 94.0
Qing 2 100.0 100.0 100.0 81.0 100.0 100.0 100.0 100.0 100.0 100.0 99.0 95.0 100.0
Quadratic 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 98.0 100.0 100.0
Quintic 2 96.0 99.0 99.0 99.0 94.0 100.0 100.0 28.0 86.0 99.0 90.0 100.0 42.0
Rana 2 100.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0 100.0 0.0 0.0 100.0
Rastrigin 2 44.0 73.0 100.0 7.0 80.0 87.0 100.0 100.0 16.0 99.0 91.0 100.0 100.0
Ratkowsky01 4 1.0 77.0 99.0 90.0 98.0 100.0 1.0 18.0 73.0 91.0 0.0 99.0 45.0
Ratkowsky02 3 94.0 100.0 100.0 100.0 95.0 100.0 43.0 95.0 92.0 99.0 98.0 100.0 97.0
ReduxSum 2 100.0 100.0 100.0 87.0 99.0 100.0 100.0 100.0 96.0 100.0 100.0 100.0 100.0
Ripple01 2 26.0 0.0 95.0 3.0 77.0 69.0 29.0 61.0 2.0 45.0 39.0 1.0 0.0
Ripple25 2 80.0 96.0 100.0 17.0 74.0 88.0 100.0 100.0 19.0 96.0 61.0 97.0 100.0
Rosenbrock 2 100.0 100.0 100.0 99.0 100.0 100.0 99.0 100.0 100.0 100.0 82.0 100.0 100.0
RosenbrockDisc 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 0.0 100.0 100.0
RosenbrockModified 2 57.0 19.0 69.0 13.0 28.0 37.0 100.0 92.0 15.0 86.0 68.0 66.0 100.0
RotatedEllipse01 2 100.0 100.0 100.0 90.0 100.0 100.0 100.0 100.0 99.0 100.0 99.0 100.0 100.0
RotatedEllipse02 2 100.0 100.0 100.0 90.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Salomon 2 27.0 97.0 37.0 0.0 37.0 49.0 9.0 47.0 3.0 21.0 84.0 0.0 1.0
Sargan 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 98.0 99.0 100.0 100.0
SawtoothXY 2 100.0 99.0 100.0 54.0 93.0 100.0 100.0 98.0 70.0 100.0 99.0 100.0 100.0
Schaffer01 2 11.0 40.0 86.0 0.0 67.0 98.0 6.0 61.0 6.0 83.0 100.0 1.0 5.0
Schaffer02 2 13.0 40.0 100.0 1.0 75.0 100.0 9.0 100.0 30.0 92.0 100.0 6.0 6.0
Schaffer03 2 17.0 49.0 100.0 3.0 79.0 98.0 3.0 94.0 25.0 88.0 93.0 0.0 21.0
Schaffer04 2 13.0 34.0 99.0 1.0 78.0 94.0 2.0 95.0 14.0 79.0 85.0 0.0 4.0
SchmidtVetters 3 100.0 0.0 100.0 0.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0
Schwefel01 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 97.0 100.0 100.0 100.0
Schwefel02 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.0 100.0 100.0
Schwefel04 2 100.0 100.0 100.0 100.0 91.0 100.0 100.0 100.0 100.0 100.0 98.0 100.0 100.0
Schwefel06 2 99.0 90.0 100.0 100.0 100.0 100.0 100.0 6.0 100.0 57.0 96.0 100.0 92.0
Schwefel20 2 95.0 100.0 100.0 100.0 100.0 100.0 100.0 22.0 100.0 100.0 92.0 100.0 90.0
Schwefel21 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 18.0 100.0 89.0 98.0 100.0 89.0
Schwefel22 2 97.0 100.0 100.0 99.0 100.0 100.0 100.0 22.0 100.0 99.0 99.0 100.0 82.0
Schwefel26 2 100.0 9.0 100.0 32.0 70.0 97.0 100.0 100.0 34.0 96.0 96.0 100.0 100.0
Schwefel36 2 100.0 100.0 100.0 0.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Shekel05 4 89.0 55.0 59.0 55.0 69.0 71.0 77.0 76.0 43.0 98.0 54.0 68.0 95.0
Shekel07 4 73.0 63.0 57.0 57.0 71.0 77.0 73.0 75.0 44.0 96.0 53.0 70.0 89.0
Shekel10 4 61.0 61.0 46.0 55.0 65.0 77.0 81.0 60.0 30.0 97.0 52.0 82.0 79.0
Shubert01 2 95.0 96.0 100.0 20.0 82.0 99.0 86.0 99.0 56.0 100.0 95.0 60.0 100.0
Shubert03 2 92.0 75.0 100.0 7.0 81.0 100.0 93.0 99.0 27.0 100.0 97.0 9.0 100.0
Shubert04 2 94.0 80.0 100.0 16.0 91.0 99.0 96.0 100.0 50.0 100.0 97.0 72.0 100.0
Simpleton 10 100.0 100.0 100.0 100.0 100.0 0.0 0.0 100.0 0.0 100.0 100.0 0.0 0.0
SineEnvelope 2 9.0 0.0 18.0 0.0 15.0 12.0 1.0 17.0 2.0 49.0 47.0 0.0 3.0
Sinusoidal 2 99.0 11.0 97.0 32.0 87.0 97.0 100.0 100.0 65.0 100.0 96.0 100.0 94.0
SixHumpCamel 2 100.0 99.0 100.0 86.0 92.0 100.0 100.0 100.0 79.0 100.0 97.0 100.0 100.0
Sodp 2 100.0 100.0 100.0 99.0 100.0 100.0 100.0 100.0 100.0 100.0 99.0 100.0 100.0
Sphere 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 98.0 100.0 100.0
SphericalSinc 2 93.0 15.0 99.0 16.0 94.0 100.0 100.0 100.0 64.0 100.0 98.0 100.0 100.0
Spike 2 57.0 80.0 99.0 1.0 62.0 78.0 45.0 97.0 1.0 76.0 16.0 25.0 81.0
Step01 2 100.0 0.0 100.0 93.0 91.0 100.0 100.0 100.0 85.0 88.0 100.0 100.0 72.0
Step02 2 100.0 0.0 100.0 92.0 91.0 100.0 100.0 100.0 86.0 100.0 98.0 100.0 67.0
Step03 2 100.0 1.0 100.0 100.0 96.0 100.0 100.0 100.0 99.0 100.0 97.0 100.0 64.0
Stochastic 2 27.0 0.0 99.0 44.0 82.0 95.0 100.0 0.0 0.0 0.0 26.0 0.0 0.0
StretchedV 2 100.0 100.0 100.0 100.0 97.0 98.0 100.0 100.0 100.0 100.0 98.0 68.0 100.0
StyblinskiTang 2 100.0 64.0 100.0 58.0 96.0 100.0 100.0 100.0 67.0 100.0 100.0 100.0 100.0
TestTubeHolder 2 63.0 33.0 97.0 7.0 76.0 64.0 100.0 99.0 10.0 99.0 54.0 58.0 100.0
ThreeHumpCamel 2 100.0 99.0 98.0 80.0 95.0 99.0 100.0 100.0 72.0 99.0 100.0 100.0 100.0
Thurber 7 0.0 0.0 41.0 59.0 0.0 0.0 0.0 0.0 0.0 22.0 0.0 6.0 0.0
Treccani 2 100.0 97.0 100.0 99.0 98.0 100.0 100.0 100.0 93.0 100.0 99.0 100.0 100.0
Trefethen 2 8.0 7.0 35.0 0.0 8.0 31.0 6.0 22.0 2.0 51.0 19.0 0.0 3.0
Trid 6 99.0 100.0 100.0 100.0 100.0 100.0 8.0 80.0 66.0 90.0 15.0 100.0 83.0
TridiagonalMatrix 2 100.0 100.0 100.0 100.0 95.0 100.0 100.0 100.0 99.0 100.0 100.0 100.0 100.0
Trigonometric01 2 99.0 88.0 100.0 76.0 83.0 99.0 100.0 97.0 57.0 91.0 99.0 100.0 99.0
Trigonometric02 2 98.0 100.0 100.0 68.0 89.0 100.0 100.0 100.0 51.0 75.0 97.0 100.0 0.0
Tripod 2 89.0 58.0 73.0 71.0 75.0 76.0 100.0 23.0 61.0 99.0 69.0 74.0 85.0
Tsoulos 2 65.0 98.0 100.0 22.0 80.0 92.0 100.0 100.0 27.0 100.0 95.0 96.0 100.0
Ursem01 2 100.0 83.0 100.0 80.0 91.0 100.0 100.0 100.0 98.0 100.0 99.0 100.0 100.0
Ursem03 2 80.0 67.0 100.0 26.0 87.0 99.0 100.0 12.0 50.0 100.0 98.0 97.0 68.0
Ursem04 2 100.0 100.0 100.0 75.0 99.0 100.0 100.0 39.0 98.0 99.0 100.0 100.0 100.0
UrsemWaves 2 100.0 100.0 100.0 0.0 0.0 0.0 100.0 0.0 0.0 100.0 100.0 0.0 100.0
VenterSobiezcczanskiSobieski 2 100.0 42.0 100.0 31.0 87.0 95.0 100.0 100.0 31.0 98.0 98.0 100.0 95.0
Vincent 2 100.0 80.0 100.0 90.0 58.0 63.0 90.0 94.0 87.0 100.0 95.0 90.0 100.0
Watson 6 30.0 99.0 100.0 100.0 100.0 0.0 0.0 44.0 0.0 86.0 0.0 71.0 66.0
Wavy 2 28.0 39.0 100.0 7.0 84.0 96.0 94.0 100.0 36.0 99.0 98.0 100.0 100.0
WayburnSeader01 2 100.0 100.0 100.0 98.0 90.0 99.0 100.0 99.0 89.0 99.0 95.0 99.0 100.0
WayburnSeader02 2 100.0 100.0 100.0 87.0 100.0 100.0 100.0 97.0 100.0 100.0 96.0 99.0 100.0
WayburnSeader03 2 100.0 100.0 100.0 85.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Weierstrass 2 98.0 0.0 100.0 86.0 88.0 100.0 100.0 0.0 72.0 45.0 0.0 100.0 0.0
Whitley 2 12.0 24.0 22.0 2.0 28.0 37.0 5.0 19.0 3.0 13.0 24.0 0.0 5.0
Wolfe 3 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
XinSheYang01 2 71.0 0.0 100.0 61.0 84.0 99.0 100.0 9.0 6.0 38.0 51.0 1.0 0.0
XinSheYang02 2 98.0 97.0 94.0 22.0 72.0 90.0 100.0 14.0 39.0 100.0 94.0 100.0 68.0
XinSheYang03 2 7.0 15.0 28.0 4.0 3.0 9.0 37.0 66.0 1.0 91.0 15.0 16.0 100.0
XinSheYang04 2 56.0 2.0 93.0 7.0 73.0 87.0 100.0 19.0 31.0 99.0 91.0 98.0 69.0
Xor 9 100.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0 0.0 100.0 0.0 0.0 0.0
YaoLiu04 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 19.0 100.0 94.0 97.0 100.0 94.0
YaoLiu09 2 56.0 79.0 100.0 7.0 81.0 93.0 100.0 100.0 26.0 100.0 93.0 99.0 100.0
Zacharov 2 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.0 99.0 100.0 100.0
Zagros 2 16.0 0.0 47.0 0.0 67.0 35.0 0.0 0.0 1.0 0.0 0.0 68.0 0.0
ZeroSum 2 0.0 0.0 97.0 79.0 98.0 99.0 96.0 0.0 98.0 46.0 0.0 100.0 0.0
Zettl 2 99.0 100.0 100.0 99.0 100.0 100.0 100.0 100.0 99.0 99.0 98.0 100.0 100.0
Zimmerman 2 92.0 0.0 45.0 43.0 45.0 54.0 0.0 0.0 10.0 0.0 43.0 40.0 0.0
Zirilli 2 100.0 99.0 100.0 83.0 98.0 97.0 100.0 100.0 83.0 100.0 100.0 100.0 100.0