Convergence Indicators

Overview

The Convergence Indicators widget tracks how well the current Pareto front covers the objective space as trials accumulate. A dropdown lets you switch among four multi-objective convergence indicators:

Indicator	Direction	Description
Hypervolume (HV)	Higher is better	Volume of objective space dominated by the Pareto front and bounded by a reference point.
IGD+	Lower is better	Modified Inverted Generational Distance — average distance from a reference set to the nearest Pareto-front point.
ε-indicator	Lower is better	Smallest ε such that every reference-set point is ε-dominated by some Pareto-front point.
R2	Lower is better	Utility-based indicator measuring expected gap from an ideal reference set.

For how to choose among them, see "How to choose the indicator"; for their definitions, see "Indicator definitions" at the end.

Note: These indicators are defined only for multi-objective studies with ≥ 2 objectives.

Comparison Studies

When comparison studies are added, how series are made comparable depends on the indicator:

Indicator	Behavior when comparing studies
IGD+ / ε-indicator / R2	All series are evaluated against a shared reference set, computed from the union of all studies, and normalized to [0, 1]. This makes them directly comparable on the same chart.
Hypervolume	Instead of a reference set, all series share a common reference point, defined by the union's nadir plus a 10% margin or the manual override below. Values are not normalized to [0, 1] and remain in raw hypervolume units.

Operations

Pan, zoom, and reset follow the standard 2D chart controls.

Item	Description
Indicator selector	Use the dropdown at the top to switch indicators. The chart recomputes automatically.
Reference point (Hypervolume only)	Override the auto-computed reference point for each objective, based on the nadir plus a 10% margin.

How to Read

If the curve improves monotonically, new trials are continuing to expand the Pareto front, which indicates that the multi-objective optimization is making steady progress. A pattern with rapid improvement early on followed by smaller and smaller changes is typical when the optimizer finds good solutions at an early stage and then gradually slows as it approaches convergence.

By contrast, a large jump in the indicator means that a newly found solution significantly expanded the Pareto front. If the curve becomes flat and no longer improves, the Pareto front quality has likely plateaued. This may indicate convergence, but you should also confirm that increasing the number of trials or trying a different sampler or configuration does not significantly change the result.

Although four indicators are available, comparing their absolute values against each other is not meaningful. It is more important to compare multiple studies using the same indicator.

How to choose the indicator

All four indicators express "how good the Pareto front is" as a single scalar, but they measure different aspects and have different computational properties.

Indicator	Direction	Property	When to pick
Hypervolume	Higher is better	Dominated volume (needs only a reference point; raw units)	When you want a single volume that summarizes both convergence and spread
IGD+	Lower is better	Mean distance to a reference set (mean-based), weakly Pareto compliant	The standard indicator for the average closeness of convergence
ε-indicator	Lower is better	Worst-case distance to a reference set (max-based)	When you want to sensitively detect gaps (unreached regions) in the front
R2	Lower is better	Expected gap under weighted utility functions	When you want a cheap indicator that averages convergence across many preference directions

For a standard view of average convergence, pick IGD+ (or Hypervolume for an overall quality summary).
To strictly catch gaps (unreached regions) in the front, pick ε-indicator.
For a lightweight indicator that handles many preference directions on average, pick R2.
To overlay multiple comparison studies on one normalized chart, IGD+ / ε-indicator / R2 are normalized to [0, 1] and comparable. Hypervolume is evaluated against a shared reference point but stays in raw units (not normalized).

Indicator definitions

Hypervolume

The volume of objective space dominated by a point set $P$ and bounded above by a reference point $r$ :

$\mathrm{HV}(P; r) = \mathrm{Leb}!\left( \bigcup_{p \in P} [p, r] \right)$

The reference point is usually auto-computed as the overall nadir (worst value per objective) plus a 10% margin, and can be overridden manually on the Convergence Indicators widget.
A larger dominated volume means a better front, summarizing both convergence and spread in a single volume.
Unlike the other three, it needs no reference set — only a reference point.

IGD+

The average of the "modified distance" $d^+$ from each point of a reference set $Z$ to the nearest point in the approximation set $A$ :

$\mathrm{IGD}^+(A) = \frac{1}{|Z|} \sum_{z \in Z} \min_{a \in A} d^+(a, z), \qquad d^+(a, z) = \sqrt{\sum_j \max(a_j - z_j, 0)^2}$

$d^+$ sums only the dimensions where $a$ is worse than $z$ , so the indicator does not worsen as long as $A$ (weakly) dominates $Z$ (weakly Pareto compliant).
It is a mean-based indicator; a smaller value means better convergence.

ε-indicator

The smallest translation that makes the approximation set ε-dominate every point of the reference set:

$I_{\varepsilon+}(A, Z) = \max_{z \in Z} \min_{a \in A} \max_j (a_j - z_j)$

The outer $\max$ makes it a worst-case indicator, sensitive to gaps (unreached regions) in the front.
It can be negative when $A$ strictly dominates $Z$ , but on the Convergence Indicators widget $I_{\varepsilon+} \ge 0$ .

R2

The average, over many weighted Tchebycheff scalarizations $\max_j w_j a_j$ , of how close the approximation set gets to the ideal point:

$R2(A; W) = \frac{1}{|W|} \sum_{w \in W} \min_{a \in A} \max_j w_j, a_j$

The weight vectors $W$ are generated by a Das–Dennis simplex lattice, with each objective normalized to $[0,1]$ (ideal = origin).
Cheaper than hypervolume and sensitive to both convergence and spread.