This paper studies preference-shaped expected improvement criteria for Bayesian multiobjective optimization. We consider two indicator families which are often used for similar algorithmic purposes, but which are geometrically different. The hypervolume indicator is based on a dystopian reference point and measures dominated volume in objective space. The R2 indicator is based on a utopian point and evaluates approximation sets through weighted Tchebycheff scalarization envelopes. The purpose of the paper is to make precise which preference transformations preserve exact computation, Pareto compatibility, and monotonicity properties, and which transformations change the underlying geometry. On the hypervolume side, we revisit canonical EHVI through the Deng representation, formulate product-density weighted EHVI in desirability coordinates, discuss cone-based EHVI as ordinary EHVI after a linear cone transformation, and separate these cases from truncated EHVI, where variance monotonicity may fail. On the R2 side, we prove that exact integral R2 improvement is not, in general, an ordinary objective-space weighted hypervolume. The obstruction is lower-dimensional: Lebesgue-density hypervolume cannot see certain boundary contributions that Tchebycheff scalarizations still detect. We then show that exact integral R2 improvement is exactly a scalarization-space volume, namely the measure of the Tchebycheff shadow between the incumbent scalarization envelope and the reference envelope. This representation yields finite-sum ER2I algorithms for discrete R2, quadrature methods for exact integral R2, and an achievement-space Gaussian surrogate formulation in which ER2I is an integral of scalar Gaussian expected improvements.

翻译：本文研究贝叶斯多目标优化中偏好形状化的期望改进准则。我们考虑两类在类似算法目的中常被使用但几何性质不同的指标。超体积指标基于反乌托邦参考点，测量目标空间中的支配体积。R2指标基于乌托邦点，通过加权切比雪夫标量化包络评估近似集。本文旨在明确哪些偏好变换能够保持精确计算、帕累托相容性和单调性性质，以及哪些变换会改变底层几何结构。在超体积方面，我们通过邓表示重新审视经典EHVI，在期望性坐标中构建乘积密度加权EHVI，讨论基于锥的EHVI作为线性锥变换后的普通EHVI，并将这些情况与截断EHVI区分开来，后者可能违反方差单调性。在R2方面，我们证明精确积分R2改进一般不是普通目标空间加权超体积。障碍在于低维性：勒贝格密度超体积无法捕捉切比雪夫标量化仍能检测到的某些边界贡献。接着，我们证明精确积分R2改进恰好是标量化空间体积，即当前标量化包络与参考包络之间切比雪夫阴影的测度。该表示能导出离散R2的有限和ER2I算法、精确积分R2的正交方法，以及一种成就空间高斯替代公式，其中ER2I是标量高斯期望改进的积分。