Modified Distance Calculation in Generational Distance and Inverted Generational Distance

Abstract

Abstract. In this paper, we propose the use of modified distance calculation in generational distance (GD) and inverted generational distance (IGD). These performance indicators evaluate the quality of an obtained solution set in comparison with a pre-specified reference point set. Both indicators are based on the distance between a solution and a reference point. The Euclidean distance in an objective space is usually used for distance calculation. Our idea is to take into account the dominance relation between a solution and a reference point when we calculate their distance. If a solution is dominated by a reference point, the Euclidean distance is used for their distance calculation with no modification. However, if they are non-dominated with each other, we calculate the minimum distance from the reference point to the dominated region by the solution. This distance can be viewed as an amount of the inferiority of the solution (i.e., the insufficiency of its objective values) in comparison with the reference point. We demonstrate using simple examples that some Pareto non-compliant results of GD and IGD are resolved by the modified distance calculation. We also show that IGD with the modified distance calculation is weakly Pareto compliant whereas the original IGD is Pareto non-compliant.

An Empirical Assessment of the Properties of Inverted Generational Distance on Multi- and Many-Objective Optimization

“…Both I H and I 1 are Pareto-compliant [14], 5 whereas the IGD is not [9]. Despite this drawback, the IGD has become widely adopted in EMO studies of many-objective optimization problems, i.e., problems with more than three objectives.…”

“…The averaged Hausdorff distance (∆ p [11]) is a variant of IGD that attempts to alleviate some of these drawbacks, except for the lack of Paretocompliance. More recently, the IGD + has been proposed as a Pareto-compliant variant of IGD [9], being very similar to the IGD and being as robust as ∆ p to different sizes of reference fronts. The properties of the IGD + are still poorly understood and so far no investigation has been conducted about the behavior of the IGD + with respect to convergence, spread, and distribution.…”

“…Alternative versions of the IGD have been proposed in recent years [9,11] with hopes of addressing its potential drawbacks. In addition to its lack of Paretocompliance, the IGD is not strictly a distance since under some conditions it does not satisfy the triangle inequality among approximation fronts [11].…”

“…Thus, their relative quality is typically evaluated with the aid of quality metrics, also known as quality indicators [14], which provide a complete ranking. Many quality metrics have been proposed in the literature [9,[11][12][13][14], and multiple quality metrics are often used simultaneously, because each metric assigns different importance to various desirable features of approximation fronts, such as convergence, spread and distribution. These differences between metrics may lead to "disagreements", when each metric chooses different approximation fronts as the best ones.…”

Metamodel-based prediction of performance metrics for bilevel parameter tuning in MOEAs

“…Secondly, in terms of computational complexity, the calculation of I IGD requires O(m|S| · |P |) operations, while that of hypervolume, being O(|S| m−1 ), is much more expensive [4]. Finally, it is to be noted that unlike hypervolume, I IGD is not Pareto-complaint [7], i.e., I IGD (S 1 ) < I IGD (S 2 ) S 1 S 2 . Despite these differences, both I H and I IGD are equally popular and are among the most commonly used MOEA performance metrics.…”