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

Lennart Schäpermeier1, Pascal Kerschke2

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Evolutionary Computation
|October 9, 2025
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Summary
This summary is machine-generated.

This study introduces a continuous variant of the R2 indicator for multi-objective optimization. This new indicator is Pareto-compliant and computationally efficient, offering an improved alternative for assessing solution set quality.

Keywords:
Pareto compliancePerformance assessmentR2 indicatorbenchmarkingmulti-objective optimizationutility functions

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Area of Science:

  • Multi-objective optimization
  • Decision analysis
  • Computational intelligence

Background:

  • Set-based quality indicators are crucial for evaluating solutions in multi-objective optimization.
  • The traditional R2 indicator, while common, is only weakly Pareto-compliant due to distribution discretization.
  • This limitation means adding better solutions may not always improve the R2 score.

Purpose of the Study:

  • To reinvestigate the R2 indicator using a continuous, uniform distribution of utility functions.
  • To develop a strictly Pareto-compliant version of the R2 indicator.
  • To provide efficient computational methods for this improved indicator.

Main Methods:

  • Analysis of R2 indicator properties under a continuous uniform distribution of Tchebycheff utility functions.
  • Development of an O(NlogN) algorithm for computing the indicator in bi-objective problems.
  • Implementation of incremental update procedures for adding/removing solutions.

Main Results:

  • The continuous R2 indicator is proven to be strictly Pareto-compliant.
  • Efficient computational procedures are established for bi-objective problems.
  • Incremental updates significantly reduce recomputation costs when solution sets change.

Conclusions:

  • The continuous R2 indicator offers a theoretically sound and practically efficient performance metric.
  • It serves as a promising alternative to existing Pareto-compliant metrics like the hypervolume indicator.
  • This work advances the field of performance assessment in multi-objective optimization.