  Evolutionary algorithms (EAs) are the preferred method for solving black-box
multi-objective optimization problems, but when gradients of the objective
functions are available, it is not straightforward to exploit these
efficiently. By contrast, gradient-based optimization is well-established for
single-objective optimization. A single-objective reformulation of the
multi-objective problem could therefore offer a solution. Of particular
interest to this end is the recently introduced uncrowded hypervolume (UHV)
indicator, which takes into account dominated solutions. In this work, we show
that the gradient of the UHV can often be computed, which allows for a direct
application of gradient ascent algorithms. We compare this new approach with
two EAs for UHV optimization as well as with one gradient-based algorithm for
optimizing the well-established hypervolume. On several bi-objective
benchmarks, we find that gradient-based algorithms outperform the tested EAs by
obtaining a better hypervolume with fewer evaluations whenever exact gradients
of the multiple objective functions are available and in case of small
evaluation budgets. For larger budgets, however, EAs perform similarly or
better. We further find that, when finite differences are used to approximate
the gradients of the multiple objectives, our new gradient-based algorithm is
still competitive with EAs in most considered benchmarks. Implementations are
available at https://github.com/scmaree/uncrowded-hypervolume.
