  We study the approximation of general multiobjective optimization problems
with the help of scalarizations. Existing results state that multiobjective
minimization problems can be approximated well by norm-based scalarizations.
However, for multiobjective maximization problems, only impossibility results
are known so far. Countering this, we show that all multiobjective optimization
problems can, in principle, be approximated equally well by scalarizations. In
this context, we introduce a transformation theory for scalarizations that
establishes the following: Suppose there exists a scalarization that yields an
approximation of a certain quality for arbitrary instances of multiobjective
optimization problems with a given decomposition specifying which objective
functions are to be minimized / maximized. Then, for each other decomposition,
our transformation yields another scalarization that yields the same
approximation quality for arbitrary instances of problems with this other
decomposition. In this sense, the existing results about the approximation via
scalarizations for minimization problems carry over to any other objective
decomposition -- in particular, to maximization problems -- when suitably
adapting the employed scalarization.
  We further provide necessary and sufficient conditions on a scalarization
such that its optimal solutions achieve a constant approximation quality. We
give an upper bound on the best achievable approximation quality that applies
to general scalarizations and is tight for the majority of norm-based
scalarizations applied in the context of multiobjective optimization. As a
consequence, none of these norm-based scalarizations can induce approximation
sets for optimization problems with maximization objectives, which unifies and
generalizes the existing impossibility results concerning the approximation of
maximization problems.
