abstract = "Hypervolume (HV) has become one of the most popular
indicators to assess the quality of Pareto front
approximations. However, the best algorithm for
computing these values has a computational complexity
of O(Nk/3polylog(N)) for N candidate solutions and k
objectives. we propose a regression-based approach to
learn new mathematical expressions to approximate the
HV value and improve at the same time their
computational efficiency. In particular, Genetic
Programming is used as the modelling technique, because
it can produce compact and efficient symbolic models.
To evaluate this approach, we exhaustively measure the
deviation of the new models against the real HV values
using the DTLZ and WFG benchmark suites. We also test
the new models using them as a guiding mechanism within
the indicator-based algorithm SMS-EMOA. The results are
very consistent and promising since the new models
report very low errors and a high correlation for
problems with 3, 4, and 5 objectives. What is more
striking is the execution time achieved by these
models, which in a direct comparison against standard
HV calculation achieved extremely high speedups of
close to 100X for a single front and over 1000X for all
the HV contributions in a population, speed-ups reach
over ten fold in full runs of SMS-EMOA compared with
the standard Monte Carlo approximations of the HV,
particularly for large population sizes. Finally, the
evolved models generalize across multiple complex
problems, using only two problems to train the problems
from the DTLZ benchmark and performing efficiently and
effectively on all remaining DTLZ and WFG benchmark
problems.",