F_{opt} Known  X_{opt} Known  Difficulty 

Yes  Yes  Easy 
The MMTFG test function generator of global optimization test functions is almost as famous as the GKLS one. It is described in the paper A framework for generating tunable test functions for multimodal optimization. I have taken the original C code (available at http://www.ronkkonen.com/generator/ ), and I have instructed the benchmark runner (that is written in Python) to communicate with the C driver via input files  the default for the MMTFG generator, as my Cython skills were not good enough to wrap the code as it is.
The acronym MMTFG obviously stands for MultiModal Test Function Generator, as the article above describes it.
Even though there are no code conversions involved, I still like to check that things add up correctly while using a test functions generator. I then have compared the results of the original C code and of my bridge to it ad they are exactly the same: this is easy to see by comparing Figure 2 (a) and Figure 2 (b) in the paper above with Figure 4.1 below. Thr first picture shows a MMTFG quadratic family function from a twodimensional test functions with 10 spherical, global minima. The second figure has 10 global and 100 local rotated ellipsoidal minima such that the local minima points have fitness value in range . Globally optimal value is always 1.
The MMTFG generator can be used to generate multimodal test functions for minimization, tunable with parameters to allow generation of landscape characteristics that are specifically designed for evaluating multimodal optimization algorithms by their ability to locate multiple optima. At the moment, three families of functions exist, but the generator is easily expandable and new families may be added in future. The current families are cosine, quadratic and common families.
The cosine family samples two cosine curves together, of which one defines global minima and another adds local minima. The basic internal structure is regular: all minima are of similar size and shape and located in rows of similar distance from each other. The function family is defined by:
where , is the dimensionality, the parameters and are vectors of positive integers which define the number of global and local minima for each dimension, and defines the amplitude of the sampling function (depth of the local minima). The generator allows the function to be rotated to a random angle and use Bezier curves to stretch each dimension independently to decrease the regularity.
The quadratic family can be used to generate completely irregular landscapes. The function is created by combining several minima generated independently. They are described as a dimensional general quadratic form, where a symmetric matrix defines the shape of each minimum. The functions in quadratic family need not be stretched or rotated, because no additional benefit would be gained by doing that to an already irregular function. However, axisaligned minima may be randomly rotated by rotating the matrix with a matrix is a randomly generated angle preserving orthogonal linear transformation such as:
The functions in the quadratic family are then calculated by:
where , defines the location and the fitness value of a minimum point for the i’th minimum. is the number of minima.
The hump family implements the generic hump functions family proposed by Singh and Deb (2006). Like the quadratic family, the hump functions allow irregular landscapes to be generated and the number of minima to be defined independently of the dimensionality. The placement of minima is chosen randomly. Each minimum is defined by:
where , is the value of the i th minimum, is the Euclidean distance between and the center of the i th minimum, defines the basin radius and the shape of the i th minimum slope.
My approach in building the test functions using the MMTFG generator has been family dependent, and in particular:
For all families, the number of dimensions ranges from 2 to 4
Armed with all these combinations  and please keep in mind that some combinations, especially in the quadratic family, are invalid due to internal constraints in the test function generator  I have generated a total of 981 benchmark functions divided as presented in Table 4.1:
Family  N = 2  N = 3  N = 4  Total 

Cos  108  108  108  324 
Hump  96  96  96  288 
Quad  108  126  135  369 
Total  312  330  339  981 
A few examples of 2D benchmark functions created with the MMTFG generator can be seen in Figure 4.2.
Table 4.2 below shows the overall success of all Global Optimization algorithms, considering every benchmark function, for a maximum allowable budget of .
As we can see  and in contrast with the GKLS and GlobOpt test function generators  the level of difficulty of the MMTFG test suite is much lower, as the best solver ( MCS) solves up to 77% of all the problems with less than 300 function evaluations. Second comes the BiteOpt algorithm, but many of the SciPy solvers are up there too  BasinHopping, DualAnnealing and SHGO provide excellent performances on this test suite.
Note
The reported number of functions evaluations refers to successful optimizations only.
Optimization Method  Overall Success (%)  Functions Evaluations 

AMPGO  60.24%  461 
BasinHopping  72.07%  319 
BiteOpt  72.27%  515 
CMAES  52.19%  570 
CRS2  49.85%  1,007 
DE  32.93%  1,396 
DIRECT  59.43%  419 
DualAnnealing  66.97%  313 
LeapFrog  43.93%  278 
MCS  77.06%  287 
PSWARM  19.57%  890 
SCE  24.57%  958 
SHGO  69.22%  295 
These results are also depicted in Figure 4.3, which shows that MCS is the betterperforming optimization algorithm, followed by BiteOpt and BasinHopping.
Pushing the available budget to a very generous , the results show MCS taking the lead on other solvers (about 6% more problems solved compared to the second best, BiteOpt). The results are also shown visually in Figure 4.4.
Optimization Method  Overall Success (%)  Functions Evaluations 

AMPGO  75.84%  1,359 
BasinHopping  77.68%  518 
BiteOpt  78.90%  887 
CMAES  66.77%  1,449 
CRS2  61.06%  1,422 
DE  71.97%  2,880 
DIRECT  64.42%  746 
DualAnnealing  71.25%  473 
LeapFrog  43.93%  278 
MCS  85.52%  726 
PSWARM  67.69%  2,454 
SCE  35.47%  2,018 
SHGO  78.70%  679 
All the solvers except for LeapFrog and SCE are able to solve more than 60% of the problems if given a generous budget as we did above at .
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, semitransparent 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 4.5.
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 4.6 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.
A few obvious conclusions we can draw from these pictures are:
Since I used the MMTFG test suite to generate test functions with dimensionality ranging from 2 to 4, it is interesting to take a look at the solvers performances as a function of the problem dimensionality. Of course, in general it is to be expected that for larger dimensions less problems are going to be solved  although it is not always necessarily so as it also depends on the function being generated. Results are shown in Table 4.4 .
Solver  N = 2  N = 3  N = 4  Overall 

AMPGO  76.9  63.6  41.6  60.7 
BasinHopping  83.3  71.2  62.5  72.4 
BiteOpt  84.6  71.5  61.7  72.6 
CMAES  68.9  50.0  38.9  52.6 
CRS2  74.4  50.6  26.5  50.5 
DE  77.9  24.2  0.0  34.0 
DIRECT  78.2  58.2  43.4  59.9 
DualAnnealing  86.9  61.5  54.0  67.5 
LeapFrog  57.7  46.4  28.9  44.3 
MCS  92.6  76.1  63.7  77.5 
PSWARM  33.7  10.6  15.3  19.9 
SCE  39.1  18.8  16.8  24.9 
SHGO  88.8  68.5  51.9  69.7 
Figure 4.7 shows the same results in a visual way.
What we can infer from the table and the figure is that, for lower dimensionality problems (), MCS, SHGO, BasinHopping and BiteOpt solve the vast majority of problems (90%). For higher dimensionality problems (), all those solvers keep a very good performance, although it is to be expected that the number of problems solved decreases with . A dramatic drop in ability to find global optima can be seen for the DE algorithm, which crashes from 78% success rate at to 0% success for .
Pushing the available budget to a very generous , the results show MCS solving in essence all the problems at low dimensionality (), closely followed by SHGO, BasinHopping and BiteOpt with more than 90% of global minima found. For the highest dimensionality in this test suite (), we can of course observe the resurgence of DE which goes up to 55% success rate.
The results for the benchmarks at are displayed in Table 4.5 and Figure 4.8.
Solver  N = 2  N = 3  N = 4  Overall 

AMPGO  93.3  74.8  60.8  76.3 
BasinHopping  90.4  75.8  67.8  78.0 
BiteOpt  92.3  77.6  67.8  79.2 
CMAES  78.8  68.5  54.0  67.1 
CRS2  75.0  61.5  47.8  61.4 
DE  87.5  74.2  55.5  72.4 
DIRECT  87.2  59.7  48.1  65.0 
DualAnnealing  89.7  66.1  59.3  71.7 
LeapFrog  57.7  46.4  28.9  44.3 
MCS  98.7  85.5  73.5  85.9 
PSWARM  83.3  62.1  58.7  68.1 
SCE  48.1  30.0  29.2  35.8 
SHGO  95.5  81.2  60.8  79.2 
Figure 4.8 shows the same results in a visual way.
For this specific test suite, it is interesting to see if any particular family of functions is easier or harder to solve compared to the others, across all optimization algorithms. This type of results is shown in Table 4.6 and Table 4.7 below, for and maximum budget of function evaluations respectively.


What seems clear from this tables is that, no matter whether our budget is relatively limited at or very generous at , the Cosine family appears to be the toughest set to crack, while the Quadratic family is generally much easier to solve, for almost all the global optimization algorithms considered.
This can also be seen in the heatmap presented in Figure 4.9.