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We study the approximation of general multiobjective optimization problemswith the help of scalarizations. Existing results state that multiobjectiveminimization problems can be approximated well by norm-based scalarizations.However, for multiobjective maximization problems, only impossibility resultsare known so far. Countering this, we show that all multiobjective optimizationproblems can, in principle, be approximated equally well by scalarizations. Inthis context, we introduce a transformation theory for scalarizations thatestablishes the following: Suppose there exists a scalarization that yields anapproximation of a certain quality for arbitrary instances of multiobjectiveoptimization problems with a given decomposition specifying which objectivefunctions are to be minimized / maximized. Then, for each other decomposition,our transformation yields another scalarization that yields the sameapproximation quality for arbitrary instances of problems with this otherdecomposition. In this sense, the existing results about the approximation viascalarizations for minimization problems carry over to any other objectivedecomposition -- in particular, to maximization problems -- when suitablyadapting the employed scalarization.  We further provide necessary and sufficient conditions on a scalarizationsuch that its optimal solutions achieve a constant approximation quality. Wegive an upper bound on the best achievable approximation quality that appliesto general scalarizations and is tight for the majority of norm-basedscalarizations applied in the context of multiobjective optimization. As aconsequence, none of these norm-based scalarizations can induce approximationsets for optimization problems with maximization objectives, which unifies andgeneralizes the existing impossibility results concerning the approximation ofmaximization problems.

Using Scalarizations for the Approximation of Multiobjective Optimization Problems: Towards a General Theory