Robust¶

F_opt Known	X_opt Known	Difficulty
Yes	Yes	Medium

This is the last benchmark of the series, and you will forgive me if I stop after this... The Robust test suite is actually composed of 5 different kind-of analytical test function generators, containing deceptive, multimodal, flat functions depending on the settings. The theory behind this test suite is described in the paper https://www.sciencedirect.com/science/article/pii/S0020025515006301 . Matlab source code is available at http://www.alimirjalili.com/RO.html , I simply converted it to Python.

Methodology¶

The actual objective functions created by this test function generator depend on which generator is actually chosen - 5 of them are available.

For generator I:

$f(x) = \frac{1}{\sqrt {2 \pi}}e^{-0.5 \left ( \frac{ \sum_{i=1}^{n} (x_i - 1.5)^2}{0.5} \right )^2} + \frac{2}{\sqrt {2 \pi}}e^{-0.5 \left ( \frac{\sum_{i=1}^{n} (x_i - 0.5)^2}{\alpha} \right )^2}$

Where $\alpha$ indicates the robustness of the global optimum by narrowing or widening it. The robustness of the global optimum is decreased proportionally to the value of the parameter $\alpha$ .
For generator II:

$f(x) = -G(x) \cdot H(x_1) \cdot H(x_2) + \omega$

Where:

$H(x) = \frac{e^{-2x^2} \sin \left [ 2 \pi \lambda \left (x + \frac{\pi}{4 \lambda} \right ) \right ] - x^{\beta}} {3} + 0.5$

$G(x) = 1 + 10 \frac{\sum_{i=3}^{N} x_i}{N}$

This benchmark generator allows generating $(\lambda + 1)^2$ number of local optima through the search space. The search space becomes more challenging as $\lambda$ increases.
For generator III:

$f(x) = (H(x_1) + H(x_2)) \cdot G(x) - 1$

Where:

$H(x) = \frac{1}{2} - 0.3 e^{ -\left ( \frac{x-0.2}{0.004}\right )^2}- 0.5 e^{ -\left ( \frac{x-0.5}{0.05}\right )^2} - 0.3 e^{ -\left ( \frac{x-0.8}{0.004}\right )^2} + \sin(\pi x)$

$G(x) = \sum_{i=3}^{N} 50 x_i^2 + 1$

This benchmark generator generates deceptive test functions.
For generator IV:

$f(x) = (H(x_1) + H(x_2)) \cdot G(x) - 2$

Where:

$H(x) = 1.2 - 0.2 e^{ -\left ( \frac{x-0.95}{0.03}\right )^2}- 0.2 e^{ -\left ( \frac{x-0.05}{0.01}\right )^2}$

$G(x) = \sum_{i=3}^{N} 50 x_i^2$

This benchmark generator generates flat test functions.

For the Robust benchmark, I have created 420 test functions with dimensionality ranging from 2 to 6.

A few examples of 2D benchmark functions created with the Robust generator can be seen in Figure 16.1.

Robust Function 2	Robust Function 10	Robust Function 65
Robust Function 173	Robust Function 253	Robust Function 327

General Solvers Performances¶

Table 16.1 below shows the overall success of all Global Optimization algorithms, considering every benchmark function, for a maximum allowable budget of $NF = 2,000$ .

The Robust benchmark suite is a medium-difficulty test suite: the best solver for $NF = 2,000$ is BiteOpt, with a success rate of 62.4%, followed at a distance by AMPGO and DualAnnealing.

Note

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

**Table 16.1**: Solvers performances on the Robust benchmark suite at NF = 2,000
Optimization Method	Overall Success (%)	Functions Evaluations
AMPGO	52.14%	583
BasinHopping	36.19%	651
BiteOpt	62.38%	383
CMA-ES	33.57%	768
CRS2	40.48%	892
DE	27.86%	1,146
DIRECT	45.71%	600
DualAnnealing	51.43%	569
LeapFrog	22.86%	177
MCS	45.48%	268
PSWARM	28.10%	1,457
SCE	23.10%	710
SHGO	36.43%	368

These results are also depicted in Figure 16.2, which shows that BiteOpt is the better-performing optimization algorithm, followed at a distance by AMPGO and DualAnnealing.

Optimization algorithms performances on the Robust test suite at :math:`NF = 2,000`

Figure 16.2: Optimization algorithms performances on the Robust test suite at $NF = 2,000$

Pushing the available budget to a very generous $NF = 10,000$ , the results show AMPGO snatching the top spot from BiteOpt, although the two algorithms are close. DualAnnealing and MCS trail behind at a large distance. The results are also shown visually in Figure 16.3.

**Table 16.2**: Solvers performances on the Robust benchmark suite at NF = 10,000
Optimization Method	Overall Success (%)	Functions Evaluations
AMPGO	75.95%	1,897
BasinHopping	41.67%	1,100
BiteOpt	73.10%	1,272
CMA-ES	58.10%	2,330
CRS2	51.43%	1,463
DE	59.05%	3,614
DIRECT	49.05%	771
DualAnnealing	64.76%	1,317
LeapFrog	23.81%	182
MCS	59.52%	1,268
PSWARM	51.19%	1,967
SCE	31.19%	1,726
SHGO	44.29%	1,630

Optimization algorithms performances on the Robust test suite at :math:`NF = 10,000`

Figure 16.3: Optimization algorithms performances on the Robust test suite at $NF = 10,000$

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 16.4.

Figure 16.4: Percentage of problems solved given a fixed number of function evaluations on the Robust 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 16.5 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 16.5: Percentage of problems solved given a fixed number of function evaluations on the Robust test suite

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

For this specific benchmark test suite, for very small budgets (i.e., $NF < 500$ ) MCS appears to be the most successful solver.
After that threshold, BiteOpt becomes vastly superior in terms of performances compared to all other solvers.
AMPGO manages to get on the top sopt only at very large budgets.

Dimensionality Effects¶

Since I used the Robust test suite to generate test functions with dimensionality ranging from 2 to 6, 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 16.3 .

**Table 16.3**: Dimensionality effects on the Robust benchmark suite at NF = 2,000
Solver	N = 2	N = 3	N = 4	N = 5	N = 6	Overall
AMPGO	72.6	57.1	45.2	48.8	36.9	52.1
BasinHopping	58.3	40.5	34.5	23.8	23.8	36.2
BiteOpt	91.7	54.8	57.1	54.8	53.6	62.4
CMA-ES	56.0	46.4	29.8	23.8	11.9	33.6
CRS2	63.1	52.4	40.5	32.1	14.3	40.5
DE	81.0	46.4	8.3	3.6	0.0	27.9
DIRECT	71.4	51.2	50.0	44.0	11.9	45.7
DualAnnealing	79.8	50.0	52.4	40.5	34.5	51.4
LeapFrog	45.2	27.4	10.7	15.5	15.5	22.9
MCS	67.9	59.5	35.7	33.3	31.0	45.5
PSWARM	48.8	28.6	28.6	19.0	15.5	28.1
SCE	59.5	31.0	21.4	3.6	0.0	23.1
SHGO	89.3	28.6	27.4	17.9	19.0	36.4

Figure 16.6 shows the same results in a visual way.

Percentage of problems solved as a function of problem dimension at :math:`NF = 2,000`

Figure 16.6: Percentage of problems solved as a function of problem dimension for the Robust test suite at $NF = 2,000$

What we can infer from the table and the figure is that, for pretty much all dimensionalities, BiteOpt displays much better convergence rates than other solvers, only slightly beaten for $N = 3$ by MCS.

Pushing the available budget to a very generous $NF = 10,000$ shows SHGO scoring a perfect 100% success rate for low dimensionality problems ( $N = 2$ ) while AMPGO shows extremely high rates of success for higher dimensionalities.

The results for the benchmarks at $NF = 10,000$ are displayed in Table 16.4 and Figure 16.7.

**Table 16.4**: Dimensionality effects on the Robust benchmark suite at NF = 10,000
Solver	N = 2	N = 3	N = 4	N = 5	N = 6	Overall
AMPGO	86.9	82.1	75.0	69.0	66.7	76.0
BasinHopping	63.1	39.3	48.8	29.8	27.4	41.7
BiteOpt	94.0	72.6	81.0	61.9	56.0	73.1
CMA-ES	64.3	63.1	60.7	52.4	50.0	58.1
CRS2	61.9	56.0	58.3	46.4	34.5	51.4
DE	83.3	81.0	54.8	46.4	29.8	59.0
DIRECT	72.6	51.2	50.0	48.8	22.6	49.0
DualAnnealing	85.7	61.9	63.1	59.5	53.6	64.8
LeapFrog	48.8	27.4	10.7	16.7	15.5	23.8
MCS	85.7	79.8	51.2	41.7	39.3	59.5
PSWARM	70.2	66.7	59.5	33.3	26.2	51.2
SCE	69.0	41.7	39.3	4.8	1.2	31.2
SHGO	100.0	53.6	21.4	25.0	21.4	44.3

Percentage of problems solved as a function of problem dimension at :math:`NF = 10,000`

Figure 16.7: Percentage of problems solved as a function of problem dimension for the Robust test suite at $NF = 10,000$

Robust¶

Methodology¶

General Solvers Performances¶

Sensitivities on Functions Evaluations Budget¶

Dimensionality Effects¶

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Methodology¶

General Solvers Performances¶

Sensitivities on Functions Evaluations Budget¶

Dimensionality Effects¶

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