R2 Indicator

Preference-Shaped Expected Hypervolume and R2 Improvement: Exact Computation and Monotonicity

This paper analyzes preference-shaped expected improvement criteria for Bayesian multiobjective optimization, focusing on hypervolume (EHVI) and R2 indicator families. The authors establish which preference transformations preserve exact computation, Pareto compatibility, and monotonicity, and which alter the underlying geometry. A key result is that exact integral R2 improvement is not generally an objective-space weighted hypervolume but is exactly a scalarization-space volume (Tchebycheff shadow measure), enabling new finite-sum and quadrature algorithms for ER2I. The work also provides an achievement-space Gaussian surrogate formulation reducing ER2I to an integral of scalar Gaussian expected improvements.