An interactive companion to our work on visualising the landscape of multi-objective optimisation problems. It samples a two-variable slice of a bi-objective benchmark, stratifies the sampled points by Pareto dominance, and draws the result as a draggable 3D landscape alongside the matching objective-space view.

The visualiser supports the two dominance-based landscape methods discussed in the paper. They differ only in the scalar mapped to the vertical axis — which is why the control selects a landscape type rather than an arbitrary height expression.

Two ways to read height

Landscape What the height encodes Where the optima sit
Pareto landscape The non-domination level of a point after a non-dominated sort — level 1 is the nondominated (Pareto-optimal) front, level 2 is what remains after removing level 1, and so on. A lower level is a higher altitude. Pareto-optimal points form the peaks.
Cost landscape The dominance count of a point — how many sampled points dominate it (also called the dominance ratio). A point dominated by nobody sits at the top; the count is inverted for display so optima remain on the peaks.

A note on terminology, since it is easy to conflate: Pareto landscape uses the non-domination level (the level index from non-dominated sorting), whereas the older cost landscape uses the number of dominating points. The two are related but distinct — two points can share a non-domination level yet have very different dominance counts. Earlier drafts of this page labelled the height “Pareto rank”; the correct term for the proposed method is non-domination level, and “Pareto rank / dominance count” refers specifically to the cost landscape.

For both landscapes, the level-1 / zero-dominator points — the optimal front for the sampled set — are highlighted as dark markers on the surface and emphasised in the objective-space scatter.

Benchmark problems

The ZDT, DTLZ, UF, and WFG examples are browser translations of the corresponding benchmark definitions used in the local minijmetal project. UF is shown as a two-coordinate slice because the standard UF definitions need parity groups beyond two variables. The BBOB proxy entries are lightweight browser-side stand-ins, not official COCO / bbob-biobj instances — use the CSV loader below if you generate official samples locally.

Using official BBOB-biobj data

The official COCO platform lists bbob-biobj and bbob-biobj-ext as available suites. To use those exact functions here, install the Python wrapper locally, sample a two-dimensional grid, and upload the resulting CSV with the control above (expected columns: x1, x2, f1, f2 on a complete rectangular grid).

import csv
import numpy as np
import cocoex

suite = cocoex.Suite("bbob-biobj", "instances: 1", "dimensions: 2")
problem = suite[0]  # choose another index for another BBOB-biobj problem
grid = np.linspace(-5.0, 5.0, 81)

with open("bbob_biobj_sample.csv", "w", newline="") as fh:
    writer = csv.writer(fh)
    writer.writerow(["x1", "x2", "f1", "f2"])
    for x1 in grid:
        for x2 in grid:
            f1, f2 = problem([x1, x2])
            writer.writerow([x1, x2, f1, f2])

COCO setup notes are on the COCO getting started page, and suite background is in the COCO data archive.

References

The methods visualised on this page are introduced and developed in the following works:

  1. Z. Liang, Z. Cui, and M. Li. Pareto Landscape: Visualising the Landscape of Multi-objective Optimisation Problems. In Parallel Problem Solving from Nature — PPSN XVIII, Lecture Notes in Computer Science, pp. 299–315. Springer, 2024. [PDF]  [Publisher]  DOI: 10.1007/978-3-031-70085-9_19

  2. Z. Cui, Z. Liang, and M. Li. A Problem Landscape Visualisation Method for Multi-Objective Optimisation. Mathematical and Computational Applications, 31(3):67, 2026. [PDF]  [Publisher]  DOI: 10.3390/mca31030067

  3. C. M. Fonseca. Multiobjective Genetic Algorithms with Application to Control Engineering Problems. PhD thesis, University of Sheffield, 1995. (Origin of the cost landscape / dominance-ratio visualisation.)

Supporting visualisation methods referenced for comparison in the papers above include the gradient field heatmap (Kerschke & Grimme, 2017) and PLOT (Schäpermeier, Grimme & Kerschke, 2020). For the benchmark suites: ZDT (Zitzler, Deb & Thiele, 2000), DTLZ (Deb et al., 2005), WFG (Huband et al., 2006), and BBOB-biobj (Brockhoff et al., COCO platform).

This visualiser is a teaching and exploration aid. For the precise definitions, the comparison with gradient field heatmaps and PLOT, and the discussion of benchmark characteristics, please see references 1 and 2 above.