New algorithm speeds integral R2 indicator using hypervolume box decomposition

The continuous integral R2 indicator improves upon the classical finite-weight-vector R2 indicator for multi-objective optimization, enabling more accurate performance assessment and bounded archiving. Emmerich's paper presents a novel bidirectional perspective mapping between integral R2 computation and integration over unions of anchored axis-aligned boxes. By translating the ideal point to the origin and mapping the weighted Tchebycheff envelope to the complement of an anchored-box union, the Jacobian yields a weighted volume formula that hypervolume algorithms can directly reuse.

This mapping allows existing box-decomposition approaches—traditionally used for hypervolume computation—to compute the integral R2 indicator efficiently. For two or three objectives, the complexity is O(n log n); for four objectives it is O(n^2); for higher dimensions it scales as O(n^(⌊(N-1)/2⌋+1)). This represents a significant practical improvement over previous methods. The paper also establishes lower bounds: exact computation is Ω(n log n) in the algebraic decision-tree model for any fixed N≥2, and it becomes #P-hard when N is part of the input.

The implications extend beyond optimization theory. The integral R2 indicator is used in a-posteriori multi-objective optimization, skyline selection in databases, and anytime performance assessment. By enabling faster exact computation, practitioners can now evaluate solution sets more accurately and in less time, especially for common low-dimensional problems. An accompanying Python implementation on GitHub makes the method immediately usable for researchers and engineers.