Optimized Approximation Sets of Low-dimensional Benchmark Pareto Fronts

This website contains supplementary material for the paper "Optimized Approximation Sets of Low-dimensional Benchmark Pareto Fronts" by Tobias Glasmachers, submitted to the 13th International Conference on Parallel Problem Solving from Nature (PPSN XIII), Ljubljana, 2014. [pdf]

Optimization Tool


Unzip the archive. On a Unix/Linux/MacOS system with a C++ compiler and (GNU) make type make to build everything. On Windows you are on your own, however, the two programs are fully self-contained in one source file each and should compile essentially everywhere. Invoke the programs hyp2d and hyp3d without arguments to get the help text, including examples. If something fails badly then I would appreciate a bug report (email me).

Results for 3 Objective Problems

The following tables provide optimized Pareto front approximations for the tri-objective problems DTLZ1 to DTLZ4. These correspond to the best solutions found in tables 4 and 5 (column "max") in the paper. The results were obtained by running the gradient-ascent optimizer for 10,000 trials with reference point (2, 2, 2). Most cardinalities are too small to be practical for three-objective problems, however, the setup is as identical as possible to Eckart Zitzler's webpage providing close-to-optimal sets for the bi-objective case:


Similar (and sometimes slightly better or worse) results can be obtained with the tool hyp2d that is included in the above code archive for download.

problem cardinality optimal set
DTLZ1 2 csv eps png
DTLZ1 3 csv eps png
DTLZ1 4 csv eps png
DTLZ1 5 csv eps png
DTLZ1 10 csv eps png
DTLZ1 20 csv eps png
DTLZ1 50 csv eps png
DTLZ1 100 csv eps png
DTLZ1 1000 csv eps png

problem cardinality optimal set
DTLZ2-4 2 csv eps png
DTLZ2-4 3 csv eps png
DTLZ2-4 4 csv eps png
DTLZ2-4 5 csv eps png
DTLZ2-4 10 csv eps png
DTLZ2-4 20 csv eps png
DTLZ2-4 50 csv eps png
DTLZ2-4 100 csv eps png
DTLZ2-4 1000 csv eps png