A new class of test functions for global optimization
(by B. Addis and M. Locatelli)
We propose a new class of test functions for unconstrained global optimization problems for which, however, it is a priori known that the global minimum lies in the interior of a sphere centered at the origin. The class depends on some parameters through which the difficulty of the test problems can be controlled. As a basis for future comparison, we propose a selected set of these functions, with increasing difficulty, and some computational experiments with two simple global optimization algorithms.
For a detailed description of the test functions, the parameters used, and the computational experiments please refer to the paper A new class of test functions for global optimization
Documentation and source
As a basis for future comparison seven different test functions have been considered. The parameters defining these functions and the results of the global optimization method MBH (Monotonic Basin Hopping) are reported in the following table (a link to input files is in the first column).
On each test function 1000 runs have been performed. For each function we report the number of successes over the 1000 runs and the average number of local searches per run.
MBH always solves the cases with 
. It can be seen that increasing the number of local minima (i.e. increasing parameter 
from 10 to 20) only slightly worsens the performance of MBH, while different scaling, due to random selection (within the interval ![$[10,20]$](form_8.png)
) of the 
values, is a more serious source of difficulty.
As the values 
and 
are increased, we observe a clear decrease of the performance of MBH (MBH gets trapped at a local minimum at level 2 and is unable to escape from it when this is not the global minimum). Notice that MBH reaches in a relativey fast time a local minimum at level 2 (it has been observed that MBH always stops at a local minimum at level 2 and the number of local searches per run is never very large) but then is unable to escape from it.
| Function
| n | K_i | H | L2 | L3 | succ. | av. LS |
| Test 1
| 50 | 10 | 10 | 1 | 1 | 1000 | 1517 |
| Test 2
| 50 | 20 | 10 | 1 | 1 | 1000 | 2393 |
| Test 3
| 50 | Random | 10 | 1 | 1 | 1000 | 5271 |
| Test 4
| 30 | 10 | 10 | 10 | 1 | 82 | 1444 |
| Test 5
| 30 | 10 | 10 | 25 | 1 | 30 | 1810 |
| Test 6
| 30 | 10 | 10 | 25 | 4 | 17 | 1781 |
| Test 7
| 30 | 10 | 10 | 100 | 4 | 3 | 1867 |
The authors acknowledge support from Progetto FIRB ``Ottimizzazione Non Lineare su Larga Scala''.
If you have any comment and/or request you can send them at the following addresses
b.addis at ing.unifi.it
locatell at di.unito.it
The site will be continuously updated.
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