The R2 indicator: A study of its expected improvement in case of two objectives

Abstract

By a multi-objective optimization problem (MOP) – aka vector optimization problem – we mean the problem of simultaneously optimizing a finite set of real valued functions with a common domain. The object of interest for multiobjective optimization is the so-called Pareto Front (PF).The indicator based approach in solving multi-objective optimization problems has become very popular. Indicators are used, among others, to compare the quality of approximation sets to PFs produced by an algorithm or different algorithms. Among the indicators used the R2 indicator attracted wide spread interest as it is relatively frugal in using computational resources as compared to other indicators. We will study the expected improvement of this indicator given an approximation set to the PF and given a probability density function of a predictive distribution of objective function vectors. The improvement of this indicator is defined as follows: the R2 indicator is evaluated on the given approximation set of the PF to which a point in the image of the feasible set is added and the R2 indicator is evaluated on the the given approximation set of the PF, subsequently from the former the latter is subtracted; the resulting difference is the R2-improvement of the chosen point with respect to the given approximation set. The expected improvement is the mean of the improvement over the image of the feasible set with respect to the given pdf. For 2 dimensional MOPs we derive a formula for the expected improvement with respect to a probability density function of a predictive distribution of objective function vectors.