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

TL;DR

This paper introduces a continuous, Pareto-compliant variant of the R2 indicator for bi-objective optimization, with efficient computation and incremental update capabilities, improving benchmarking accuracy and practicality.

Contribution

It proposes a new continuous R2 indicator that is Pareto-compliant and computationally efficient, with methods for incremental updates, enhancing set-based quality assessment in bi-objective optimization.

Findings

The continuous R2 indicator is Pareto-compliant.

Efficient $ ext{O}(N ext{log} N)$ computation method is developed.

Incremental update procedures are provided for dynamic solution sets.

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 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.