We investigate \emph{magnitude} as a new unary and strictly Pareto-compliant quality indicator for finite approximation sets to the Pareto front in multiobjective optimization. Magnitude originates in enriched category theory and metric geometry, where it is a notion of size or point content for compact metric spaces and a generalization of cardinality. For dominated regions in the \(\ell_1\) box setting, magnitude is close to hypervolume but not identical: it contains the top-dimensional hypervolume term together with positive lower-dimensional projection and boundary contributions. This paper gives a first theoretical study of magnitude as an indicator. We consider multiobjective maximization with a common anchor point. For dominated sets generated by finite approximation sets, we derive an all-dimensional projection formula, prove weak and strict set monotonicity on finite unions of anchored boxes, and thereby obtain weak and strict Pareto compliance. Unlike hypervolume, magnitude assigns positive value to boundary points sharing one or more coordinates with the anchor point, even when their top-dimensional hypervolume contribution vanishes. We then formulate projected set-gradient methods and compare hypervolume and magnitude on biobjective and three-dimensional simplex examples. Numerically, magnitude favors boundary-including populations and, for suitable cardinalities, complete Das--Dennis grids, whereas hypervolume prefers more interior-filling configurations. Computationally, magnitude reduces to hypervolume on coordinate projections; for fixed dimension this yields the same asymptotic complexity up to a factor \(2^d-1\), and in dimensions two and three \(Θ(n\log n)\) time. These results identify magnitude as a mathematically natural and computationally viable alternative to hypervolume for finite Pareto front approximations.

翻译：我们研究了作为多目标优化中帕累托前沿有限近似集的一种新的单元、严格帕累托兼容质量指标——\emph{大小}。大小源于丰富范畴理论和度量几何，是紧度量空间的大小或点内容概念，以及基数的推广。在\(\ell_1\)箱设置下的支配区域中，大小接近但不等同于超体积：它包含高维超体积项以及正的低维投影和边界贡献。本文首次对大小作为指标进行了理论研究。我们考虑具有共同锚点的多目标最大化问题。对于由有限近似集生成的支配集，我们推导出全维投影公式，证明了锚定箱有限并集上的弱和严格集单调性，从而获得弱和严格帕累托兼容性。与超体积不同，大小为与锚点共享一个或多个坐标的边界点赋予正值，即使其高维超体积贡献为零。然后，我们制定了投影集梯度方法，并比较了双目标和三维单纯形示例上的超体积和大小。数值上，大小倾向于包含边界的种群，在合适的基数下，倾向于完整的达斯-丹尼斯网格，而超体积更偏好内部填充配置。计算上，大小在坐标投影上简化为超体积；对于固定维数，这产生了与因子\(2^d-1\)相同渐近复杂度的超体积，在二维和三维中为\(Θ(n\log n)\)时间。这些结果将大小确定为有限帕累托前沿近似中超体积的数学上自然且计算上可行的替代方案。