It is well known that, under very weak assumptions, multiobjective optimization problems admit \((1+\varepsilon ,\dots ,1+\varepsilon )\)-approximation sets (also called \(\varepsilon \)-Pareto sets) of polynomial cardinality (in the size of the instance and in \(\frac{1}{\varepsilon }\)). While an approximation guarantee of \(1+\varepsilon \) for any \(\varepsilon >0\) is the best one can expect for singleobjective problems (apart from solving the problem to optimality), even better approximation guarantees than \((1+\varepsilon ,\dots ,1+\varepsilon )\) can be considered in the multiobjective case since the approximation might be exact in some of the objectives. Hence, in this paper, we consider partially exact approximation sets that require to approximate each feasible solution exactly, i.e., with an approximation guarantee of 1, in some of the objectives while still obtaining a guarantee of \(1+\varepsilon \) in all others. We characterize the types of polynomial-cardinality, partially exact approximation sets that are guaranteed to exist for general multiobjective optimization problems. Moreover, we study minimum-cardinality partially exact approximation sets concerning (weak) efficiency of the contained solutions and relate their cardinalities to the minimum cardinality of a \((1+\varepsilon ,\dots ,1+\varepsilon )\)-approximation set.

众所周知，在非常弱的假设下，多目标优化问题承认\((1+\varepsilon ,\dots ,1+\varepsilon )\) -近似集（也称为\(\varepsilon \) -Pareto 集）多项式基数（以实例的大小和 \(\frac{1}{\varepsilon }\)表示）。虽然对任何\(\varepsilon >0\ )的近似保证\(1+\varepsilon \ )是单目标问题的最佳预期（除了解决问题的最优性之外），但比\(( 1+\varepsilon ,\dots ,1+\varepsilon )\)可以在多目标情况下考虑，因为在某些目标中近似可能是精确的。因此，在本文中，我们考虑部分精确逼近集，要求在某些目标中精确逼近每个可行解，即逼近保证为 1，同时仍获得\(1+\varepsilon \)的保证所有其他人。我们描述了多项式基数的类型，即保证一般多目标优化问题存在的部分精确近似集。此外，我们研究了关于所包含解决方案的（弱）效率的最小基数部分精确近似集，并将它们的基数与a的最小基数相关联。\((1+\varepsilon ,\dots ,1+\varepsilon )\) - 近似集。