# R2 v2: The Pareto-compliant R2 Indicator for Better Benchmarking in Bi-objective Optimization

arXiv: https://arxiv.org/abs/2407.01504 · HTML: https://mlsanity.com/paper/2407.01504/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 22 references that are in the map.
- First posted on arXiv 1 Jul 2024.

## Builds on (most cited first, 1 of 1)

- [Using Well-Understood Single-Objective Functions in Multiobjective Black-Box Optimization Test Suites](https://mlsanity.com/paper/1604.00359/) (2016)

## Built on by (most cited first, 1 of 1)

- [BONO-Bench: A Comprehensive Test Suite for Bi-objective Numerical Optimization with Traceable Pareto Sets](https://mlsanity.com/paper/2601.16970/) (2026)

## Abstract (arXiv)

In multi-objective optimization, set-based quality indicators are a cornerstone of benchmarking and performance assessment. They capture the quality of a set of trade-off solutions by reducing it to a scalar number. One of the most commonly used set-based metrics is the R2 indicator, which describes the expected utility of a solution set to a decision-maker under a distribution of utility functions. Typically, this indicator is applied by discretizing the latter distribution, yielding a weakly Pareto-compliant indicator. In consequence, adding a nondominated or dominating solution to a solution set may -- but does not have to -- improve the indicator's value. In this paper, we reinvestigate the R2 indicator under the premise that we have a continuous, uniform distribution of (Tchebycheff) utility functions. We analyze its properties in detail, demonstrating that this continuous variant is indeed Pareto-compliant -- that is, any beneficial solution will improve the metric's value. Additionally, we provide efficient computational procedures that (a) compute this metric for bi-objective problems in mathcal O (N log N), and (b) can perform incremental updates to the indicator whenever solutions are added to (or removed from) the current set of solutions, without needing to recompute the indicator for the entire set. As a result, this work contributes to the state-of-the-art Pareto-compliant unary performance metrics, such as the hypervolume indicator, offering an efficient and promising alternative.