  Many optimization problems arising in applications have to consider several
objective functions at the same time. Evolutionary algorithms seem to be a very
natural choice for dealing with multi-objective problems as the population of
such an algorithm can be used to represent the trade-offs with respect to the
given objective functions. In this paper, we contribute to the theoretical
understanding of evolutionary algorithms for multi-objective problems. We
consider indicator-based algorithms whose goal is to maximize the hypervolume
for a given problem by distributing {\mu} points on the Pareto front. To gain
new theoretical insights into the behavior of hypervolume-based algorithms we
compare their optimization goal to the goal of achieving an optimal
multiplicative approximation ratio. Our studies are carried out for different
Pareto front shapes of bi-objective problems. For the class of linear fronts
and a class of convex fronts, we prove that maximizing the hypervolume gives
the best possible approximation ratio when assuming that the extreme points
have to be included in both distributions of the points on the Pareto front.
Furthermore, we investigate the choice of the reference point on the
approximation behavior of hypervolume-based approaches and examine Pareto
fronts of different shapes by numerical calculations.
