Dear Ms Guendouzi,

The question is not clear. However I can guess that you are asking for a method to assess your meta-heuristic in a multidimensional phenotype search space. you can google on Hypervolume metric. for example you can get a look at Bader's dissertation (http://www.tik.ee.ethz.ch/sop/publicationListFiles/bade2010a.pdf) and the school behind him. You can also see my dissertation in your institution.

Thanks for your reply Mr Boultif;

I think it is some thing like multidimensional scale (MDS) or PCA !!!!!!  by which we can plot a graph of all the solutions (multidimensional ) visited during the metaheuristic iterations.

it is a graph like : http://en.wikipedia.org/wiki/Multidimensional_scaling#mediaviewer/File:RecentVotes.svg

I recall that the purpose here is to investigate if the metaheuristics is visiting heterogeneous  solutions using statistical tools.

Thanks.

You might be interested by SOFM (Self Organising Feature Maps). Basically it is a grid of vectors (neurons) that adapt to a population of individuals (during an adaptive procedure). It's a good way to reduce the dimension of a population of individuals, to find clusters or sub-populations, and display them in a 2D (or more) map for instance. (One classical example is to make a 2D map of a few hundred beers based on an initial space of 10 individual properties per beer, and beers sharing siminarities will gather on close regions on the map).

Since you loose linearity with that, it will not give you a 'measure' of the variability of the population, so it might not be useful for you, but who knows ! 

The Star Discrepancy gives a good measure of how well a given bounded search space is explored. Recent research at Verimag proposes a way to approximate the star discrepancy for search spaces that are multidimensional boxes. You can check it out here: http://www-verimag.imag.fr/~tdang/Papers/FMSD2009.pdf (Section 4.1)

You may find helpful the tool described here (eprints.sics.se/2303/2/SICS-T--2006-11--SE.ps.Z) and available here (http://www.comp.nus.edu.sg/~stevenha/viz/). The tool maps the solution space on 2D and shows the search  area  covered by your meta-heuristic. The tool is problem-independant and shows also the search trajectory and other solution graphs.

The authors didn't show how to interpret the visual output, except for a specific problem and by considering a specific feature of the optimal region in the solution landscape.

So, in general, I think the difference between the solution and coverage areas is the measure to consider to answer your question. The sparsity of the search coverage area means it is equally searched. The dense spots are neighbor solutions (ie., intensification region). The relatively limited number of these spots / their light density is an indicator that your meta-heuristic is doing more diversification and less/weak intensifications.

• Thanks a lot !