The typical W-HypE parameter file (first argument when calling the jar file), consists of three parts:
1) In the first line, the initial seed of the random number generator is specified such as:
seed 656917351
2) The second line gives the number of samples used in each hypervolume estimation, e.g.:
nrOfSamples 10000
3) The following lines specify the used weight function where we distinguish between standard distributions (keyword 'dist') and the so-called mollifiers ('moll') which typically smoothen the other standard distributions. They always start with the keyword 'dist' or 'moll', possibly with an additional number ('dist1', 'dist2', ...) if more than one key is used. The second keyword in each line indicates the type of the distribution or mollifier and is always followed by the percentage of samples to be used for this distribution. Then, distribution-specific parameters follow:
dist type alpha distributionspecificparameters
For example: dist1 line 1.0 start 0.5 0.4 0.2 0.1 0.3 0.0 0.3 end 1.5 1.5 1.5 1.5 1.5 1.5 1.5 ws 1 we 0
moll1 sgau 1.0 dim 7 var 0.05
3a) Distributions
Distributions can be one of the following and the corresponding lines should add a number behind the keyword 'dist' if more than one distribution is used:
I) Stressing Objectives with Exponential Distributions (w_1): keyword 'exp'
dist exp 1.0 lower 0 0 0 0 0 0 0 upper 5 5 5 5 5 5 5 mu 0.1 dir 3
The values behind the keyword 'lower' indicate the lower bound $b^l$ and the values behind the keyword 'upper' indicate the upper bound $b^u$. The single value behind the keyword 'mu' is the parameter $\lambda$ in the paper and the value behind the keyword 'dir' gives objective to be stressed (parameter $s$ in the paper).
II) Preference Points with Multivariate Normal Distributions (w_2): keyword 'gau'
dist gau 1.0 mu 0.5 0.4 0.2 0.1 0.3 dir 1 1 1 1 1 sigma_eps 0.05 sigma_t 0.5
The values behind the keyword 'mu' define the mean of the multivariate Normal distribution, the values behind the keyword 'dir' the direction (parameter $t$ in the paper), the values behind the keywords 'sigma_eps' and 'sigma_t' give the corresponding parameters/variances of the distribution, see paper for details.
III) Desirability Function (w_4): keyword 'des'
dist des 1.0 0.2 20 0.4 20 0.8 20
After the percentage of samples to be used for this distribution (in the example 1.0), the line continues with the values $a$ and $b$ defined in the paper while the objective function values of $a$ and $b$ are given alternatingly: $a_1$ $b_1$ $a_2$ $b_2$ ...
In the example, $a=(0.2, 0.4, 0.8)$ and $b=(20, 20, 20)$.
IV) Preference Regions with Uniform Weights (w_5): keyword 'uni'
dist uni 1.0 lower 0.5 0.4 0.2 0.1 0.3 0.0 0.3 upper 1 1 1 1 1 1 1
The values behind the keywords 'lower' and 'upper' define the corners of the bounding box in which the samples are drawn uniformly at random.
V) Dirac Lines: keyword 'line'
dist line 1.0 start 0.5 0.4 0.2 0.1 0.3 0.0 0.3 end 1.5 1.5 1.5 1.5 1.5 1.5 1.5 ws 1 we 0
The values behind the keywords 'start' and 'line' define the start and end points of the Dirac line whereas the values behind the keywords 'ws' and 'we' give the height of the Dirac line at the start ('ws') and the end point ('we') respectively.
3b) Mollifiers
I) Standard Gaussian: keyword 'sgau'
moll sgau 1.0 dim 7 var 0.05
The value behind the keyword 'dim' indicates the objective function on which the smoothening takes place (1 <= dim <= numOfObjectives), the value behind the keyword 'var' defines the variance of the Gaussian distribution. The mean of the distribution is always 0.
II) Negative Gaussian: keyword 'ngau'
moll ngau 1.0 dim 1 var 100
The value behind the keyword 'dim' indicates the objective function on which the negative Gaussian distribution is to be applied (1 <= dim <= numOfObjectives), the value behind the keyword 'var' again the distribution's variance. The distribution mean is again always 0.
In addition, two more weight functions have been described in the paper, for which combinations of distributions and mollifiers are used:
Weighted Tchebycheff Approach (w_3)
combination of line distribution and standard Gaussian mollifier, e.g.:
dist1 line 1.0 start 0.5 0.4 0.2 0.1 0.3 0.0 0.3 end 1.5 1.5 1.5 1.5 1.5 1.5 1.5 ws 1 we 0
moll1 sgau 1.0 dim 7 var 0.05
Epsilon-Constraints Approach (w_6)
combination of uniform distribution and negative Gaussian mollifier, e.g.:
dist1 uni 1.0 lower 4200 0 upper 4200 10000
moll1 ngau 1.0 dim 1 var 100
For further reference, we refer to the paper
D. Brockhoff, J. Bader, L. Thiele, and E. Zitzler (2013): Directed Multiobjective Optimization Based on the Hypervolume Indicator. Journal of Multi-Criteria Decision Analysis, 20(5--6):291--317. DOI: 10.1002/mcda.1502