F_{opt} Known |
X_{opt} Known |
Difficulty |
---|---|---|

Yes | Yes | Hard |

The GKLS generator is a very popular benchmark generator for bound-constrained global optimization test functions. It is described in the paper Software for Generation of Classes of Test Functions with Known Local and Global Minima for Global Optimization. I have taken the original C code (available at http://netlib.org/toms/) and converted it to Python.

The conversion was not strictly necessary but I didn’t feel confident enough to modify the C code and wrap it with Cython in order to allow a two way communication between GKLS and the benchmark runner. In any case, I have compared the results of the original C code and of my translation ad they are exactly the same: this is easy to see by comparing Figure 1 in the paper above with Figure 2.1 below.

The GKLS generator is a procedure for generating three types (non-differentiable, continuously differentiable, and twice continuously differentiable) of classes of test functions with known local and global minima for multiextremal multidimensional box-constrained global optimization.

The procedure consists of defining a convex quadratic function systematically distorted by polynomials in order to introduce local minima. Each test class provided by the GKLS generator consists of 100 functions constructed randomly and is defined by the following parameters:

- Problem dimension
- Number of local minima
- Value of the global minimum
- Radius of the attraction region of the global minimizer
- Distance from the global minimizer to the vertex of the quadratic function

Since the user is given freedom in the choice of dimension, number of local minima, radius of the attraction region of the global minimizer and distance from the global minimizer to the vertex of the quadratic function, my approach in generating the test functions has been the following:

- The problem dimension ranges from 2 to 6
- For each dimension, there are three types of functions generated by GKLS: non-differentiable (
**ND**), continuously differentiable (**D1**) and twice continuously differentiable (**D2**) - For each dimension, GKLS generates 100 functions

In that way, we have 5 possible dimensions, 3 possible function types and 100 different functions per function types. This gives a set of 1,500 test functions. In order to give some variability, I also decided to play with the other free parameters of the generator, and namely:

- The number of local minima is chosen randomly between 3 and 100 for each function
- The distance from the global minimizer to the vertex of the quadratic function is chosen randomly between a very small number () and half of the domain bounds (which is )
- The radius of the attraction region of the global minimizer is chosen randomly between a very small number () and half of the distance from the global minimizer to the vertex of the quadratic function (defined above)

As an example of the differences between **ND**, **D1** and **D2** functions, Figure 2.2 below contains some 3D representations
of some of the benchmark functions in the GKLS test suite.

Note

For all test problems in the GKLS test suite, and for all dimensions, I have arbitrarily set the value of
the global minimum at . The default in the original C code is but it’s
relatively easy to change by manipulating the two “global” variables `GKLS_GLOBAL_MIN_VALUE` and `GKLS_PARABOLOID_MIN`,
as the test suite requires .

Table 2.1 below shows the overall success of all Global Optimization algorithms, considering every benchmark function, for a maximum allowable budget of .

It is clear at first sight that the GKLS benchmark suite is a much tougher cookie than the * SciPy Extended* one,
as the best solver (* MCS*) only manages to find the global minimum in 28.5% of the cases, while the best SciPy
optimization routines are trailing slightly behind at 24% for **DualAnnealing** and 23.3% for **SHGO**.

Among the solvers that pass the 20% threshold of solved problems, we can see that **DIRECT** is the “cheapest” one,
using 446 functions evaluations on average, followed by **DualAnnealing**, * MCS* and **SHGO**.

Note

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

Optimization Method | Overall Success (%) | Functions Evaluations |
---|---|---|

AMPGO | 14.13% | 651 |

BasinHopping | 15.53% | 721 |

BiteOpt | 18.60% | 628 |

CMA-ES | 7.07% | 745 |

CRS2 | 12.07% | 826 |

DE | 13.73% | 1,036 |

DIRECT | 21.33% | 446 |

DualAnnealing | 24.00% | 529 |

LeapFrog | 7.53% | 227 |

MCS | 28.53% | 610 |

PSWARM | 9.93% | 1,664 |

SCE | 12.07% | 604 |

SHGO | 23.27% | 639 |

These results are also depicted in Figure 2.3, which shows that * MCS* is the better-performing optimization algorithm,
closely followed by **DualAnnealing** and **SHGO**.

Pushing the available budget to a very generous , the results show * MCS* taking a larger lead
on other solvers (about 9% more problems solved compared to the second best, **DualAnnealing**) and **AMPGO** gaining
quite some ground compared to the lower budget . There is also a significant uptick in performances
for **PSWARM** as shown in Table 2.2 and Figure 2.4.

Optimization Method | Overall Success (%) | Functions Evaluations |
---|---|---|

AMPGO | 26.60% | 2,734 |

BasinHopping | 19.40% | 1,378 |

BiteOpt | 22.80% | 1,422 |

CMA-ES | 12.47% | 2,480 |

CRS2 | 13.73% | 1,057 |

DE | 19.40% | 1,948 |

DIRECT | 28.60% | 1,478 |

DualAnnealing | 30.07% | 1,257 |

LeapFrog | 7.53% | 227 |

MCS | 39.27% | 1,658 |

PSWARM | 19.47% | 2,036 |

SCE | 13.80% | 1,123 |

SHGO | 29.53% | 1,808 |

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 2.5.

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 2.6 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:

- For this specific benchmark test suite, if you have a very limited budget in terms of function evaluations,
then
*MCS*,**SHGO**,**DualAnnealing**and**DIRECT**are a very good choice: they solve around 15% of all problems in 500 function evaluations or less. - There is a distinct separation of performances between
*MCS*and the other solvers as soon as the number of functions evaluations raises above . - Pretty much all algorithms benefit from a larger budget (i.e., ) - obviously - but the improvement is much more evident
for solvers like
*MCS*(+10%),**AMPGO**(+13%) and**PSWARM**(+9%) compared to a budget of .

Since I used the GKLS 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 2.3 .

Solver |
N = 2 |
N = 3 |
N = 4 |
N = 5 |
N = 6 |
Overall |
---|---|---|---|---|---|---|

AMPGO | 43.7 | 16.3 | 4.7 | 4.3 | 1.7 | 14.1 |

BasinHopping | 50.3 | 18.7 | 8.3 | 0.0 | 0.3 | 15.5 |

BiteOpt | 46.7 | 16.3 | 14.3 | 8.7 | 7.0 | 18.6 |

CMA-ES | 27.3 | 5.3 | 2.3 | 0.3 | 0.0 | 7.1 |

CRS2 | 35.3 | 15.3 | 7.3 | 2.3 | 0.0 | 12.1 |

DE | 49.0 | 18.7 | 1.0 | 0.0 | 0.0 | 13.7 |

DIRECT | 65.7 | 28.3 | 10.3 | 2.0 | 0.3 | 21.3 |

DualAnnealing | 68.3 | 25.0 | 13.7 | 8.3 | 4.7 | 24.0 |

LeapFrog | 25.7 | 7.7 | 1.7 | 1.7 | 1.0 | 7.5 |

MCS | 75.0 | 42.3 | 15.3 | 5.0 | 5.0 | 28.5 |

PSWARM | 44.0 | 4.7 | 1.0 | 0.0 | 0.0 | 9.9 |

SCE | 45.0 | 10.7 | 4.3 | 0.0 | 0.3 | 12.1 |

SHGO | 72.3 | 32.7 | 10.3 | 1.0 | 0.0 | 23.3 |

Figure 2.7 shows the same results in a visual way.

What we can infer from the table and the figure is that, for lower dimensionality problems (), the * MCS*
algorithm has some advantage over other solvers, although the SciPy optimizers **SHGO** and **DualAnnealing** are not
very far behind. For higher dimensionality problems (), the performance degradations are noticeable for all solvers, and a
new winner shows up: **BiteOpt**, closely followed by * MCS* and **DualAnnealing**.

Pushing the available budget to a very generous , the results show * MCS* taking a larger lead
on other solvers, especially on problems with . For higher dimensionality problems ()
**DualAnnealing** starts shining more than any other optimizer.

The results for the benchmarks at are displayed in Table 2.4 and Figure 2.8.

Solver |
N = 2 |
N = 3 |
N = 4 |
N = 5 |
N = 6 |
Overall |
---|---|---|---|---|---|---|

AMPGO | 67.3 | 34.3 | 16.7 | 8.3 | 6.3 | 26.6 |

BasinHopping | 57.0 | 25.7 | 10.7 | 2.0 | 1.7 | 19.4 |

BiteOpt | 62.0 | 18.7 | 16.0 | 9.7 | 7.7 | 22.8 |

CMA-ES | 41.7 | 12.3 | 6.0 | 2.0 | 0.3 | 12.5 |

CRS2 | 35.7 | 17.3 | 7.7 | 4.7 | 3.3 | 13.7 |

DE | 49.3 | 20.0 | 11.7 | 11.3 | 4.7 | 19.4 |

DIRECT | 79.0 | 35.7 | 19.3 | 8.0 | 1.0 | 28.6 |

DualAnnealing | 72.0 | 33.3 | 19.0 | 14.3 | 11.7 | 30.1 |

LeapFrog | 25.7 | 7.7 | 1.7 | 1.7 | 1.0 | 7.5 |

MCS | 88.7 | 59.0 | 33.3 | 9.3 | 6.0 | 39.3 |

PSWARM | 62.7 | 20.0 | 11.3 | 3.3 | 0.0 | 19.5 |

SCE | 48.7 | 12.7 | 5.7 | 0.7 | 1.3 | 13.8 |

SHGO | 84.3 | 47.7 | 10.3 | 5.3 | 0.0 | 29.5 |