🤖 AI Summary
The R2 indicator for bi-objective optimization lacks strict Pareto compatibility—i.e., adding a dominated solution may not increase the indicator value.
Method: This paper proposes an analytical variant of R2 based on a continuous uniform Tchebycheff utility function. We theoretically prove that this continuous R2 exhibits strict Pareto compatibility for bi-objective problems: adding any non-dominated solution strictly increases the indicator, and adding any dominated solution necessarily increases it. We further devise an exact O(N log N) algorithm, enabling the first efficient and theoretically compliant unary quality assessment.
Results: Experiments show that the proposed indicator achieves evaluation performance comparable to hypervolume (HV), yet with significantly higher computational efficiency. It thus fills a critical gap in bi-objective set-quality indicators by simultaneously offering rigorous theoretical guarantees and practical scalability.
📝 Abstract
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 this distribution of utility functions, 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 an efficient computational procedure to compute this metric for bi-objective problems in $mathcal O (N log N)$. 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.