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.

翻译：我们研究借助标量化方法逼近一般多目标优化问题。现有结果表明，基于范数的标量化方法可以很好地逼近多目标最小化问题。然而对于多目标最大化问题，目前仅存在不可能性结果。针对这一现状，我们证明所有多目标优化问题原则上都能通过标量化方法获得同等质量的逼近。在此背景下，我们提出标量化的变换理论，该理论建立如下结论：假设存在某种标量化方法，能够对具有给定分解方式（即指定哪些目标函数需最小化/最大化）的多目标优化问题的任意实例产生特定质量的逼近解，那么对于任意其他分解方式，我们的变换方法都能导出另一种标量化方法，使其对此分解方式下的任意问题实例产生相同的逼近质量。在此意义上，关于最小化问题通过标量化逼近的现有结论可推广至任意其他目标分解形式——尤其是最大化问题——只需适当调整所采用的标量化方法。我们进一步给出标量化方法的最优解能达到恒定逼近质量的充要条件，并推导出一般标量化方法可实现的最佳逼近质量的上界，该上界对于多目标优化中应用的大多数基于范数的标量化方法是紧的。由此可知，这些基于范数的标量化方法均无法为具有最大化目标的多目标优化问题生成逼近集，这一结论统一并推广了现有关于最大化问题逼近的不可能性结果。