The expected R2-indicator improvement for multi-objective bayesian optimization

André Deutz, Michael Emmerich, Kaifeng Yang

Research output: Chapter in Book/Report/Conference proceedingsConference contributionpeer-review

3 Citations (Scopus)


In multi-objective Bayesian optimization, an infill criterion is an important part, as it is the indicator to evaluate how much good a new set of solutions is, compared to a Pareto-front approximation set. This paper presents a deterministic algorithm for computing the Expected R2 Indicator for bi-objective problems and studies its use as an infill criterion in Bayesian Global Optimization. The R2-Indicator was introduced in 1998 by M. Hansen and A. Jaszkiewicz for performance assessment in multi-objective optimization and is more recently also used in indicator-based multi-criterion evolutionary algorithms (IBEAs). In Bayesian Global Optimization, we propose the Expected R2-indicator Improvement (ER2I) as an infill criterion. It is defined as the expected decrease of the R2 indicator by a point that is sampled from a predictive Gaussian distribution. The ER2I can also be used as a pre-selection criterion in surrogate-assisted IBEAs. It provides an alternative to the Expected Hypervolume-Indicator Improvement (EHVI) that requires a reference point, bounding the Pareto front from above. In contrast, the ER2I works with a utopian reference point that bounds the Pareto front from below. In addition, the ER2I supports preference modelling with utility functions and its computation time grows only linearly with the number of considered weight combinations. It is straightforward to approximate the ER2I by Monte Carlo Integration, but so far a deterministic algorithm to solve the non-linear integral remained unknown. We outline a deterministic algorithm for the computation of the bi-objective ER2I with Chebychev utility functions. Moreover, we study monotonicity properties of the ER2I w.r.t. parameters of the predictive distribution and numerical simulations demonstrate fast convergence to Pareto fronts of different shapes and the ability of the ER2I Bayesian optimization to fill gaps in the Pareto front approximation.

Original languageEnglish
Title of host publicationEvolutionary Multi-Criterion Optimization - 10th International Conference, EMO 2019, Proceedings
EditorsSanaz Mostaghim, Kalyanmoy Deb, Erik Goodman, Kaisa Miettinen, Patrick Reed, Carlos A. Coello Coello, Kathrin Klamroth
PublisherSpringer-Verlag Italia Srl
Number of pages12
ISBN (Print)9783030125974
Publication statusPublished - 2019
Event10th International Conference on Evolutionary Multi-Criterion Optimization, EMO 2019 - East Lansing, United States
Duration: 10 Mar 201913 Mar 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11411 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference10th International Conference on Evolutionary Multi-Criterion Optimization, EMO 2019
Country/TerritoryUnited States
CityEast Lansing


  • Chebychev utility function
  • Expected improvement
  • Multiobjective Bayesian optimization
  • R2 indicator
  • Surrogate models


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