In multiobjective optimization, no single solution wins on every criterion. The trade-off surface appears as an envelope curve between axes, and parallel coordinates make high-dimensional trade-offs visible.
A car design wants high fuel efficiency, fast acceleration, high safety, low price, and large cargo space. No single design maximizes all five. The set of designs where no objective can be improved without worsening another is the Pareto front (Vilfredo Pareto).
Formally, a solution dominates a solution if is at least as good as on every objective, and strictly better on at least one:
A solution is Pareto-optimal if no other solution dominates it. In two-objective scatterplots, the Pareto front is a boundary curve. In parallel coordinates, it's an envelope of non-dominated polylines.
For 200 candidates with two objectives, the scatterplot traces the front along the feasible boundary. In PC, the Pareto-optimal segments form an envelope that no other segment crosses on both axes simultaneously.
Hover a point in the scatterplot or a line in the PC view to highlight the corresponding solution in both views.
Figure 1. 200 solutions with two objectives. Left: scatterplot with the Pareto front highlighted in red. Right: parallel coordinates. The same non-dominated solutions form an envelope boundary.
This is the envelope construction from Explainer 03: a family of non-dominated polylines whose envelope defines the boundary of the feasible region.
Real problems have more objectives. Consider 100 car designs across five criteria: fuel efficiency, acceleration time, safety rating, price, cargo space. Fuel efficiency and acceleration trade off; price and safety trade off.
The 5D Pareto front is a hypersurface. A scatterplot matrix would need 10 panels and still miss three-way interactions. Parallel coordinates show all five at once: Pareto-optimal designs (red) bound the polyline bundle; dominated ones (gray) stay inside.
Hover any polyline to see the design's values on all five objectives.
Figure 2. 100 car designs with 5 objectives. Red polylines are Pareto-optimal: no other design beats them on all five criteria. Gray polylines are dominated. The Pareto front forms a boundary envelope in the parallel coordinates view.
Inter-axis slopes encode trade-off rates. Steep slope from fuel efficiency to acceleration: a large speed sacrifice for a small mileage gain. Shallow slope: mild trade-off.
Dominance connects to the interior point test from Article 6: a solution is dominated iff it lies inside the cone of some Pareto-optimal solution. In parallel coordinates, the dominated polyline never rises above the dominating one on any axis.
Selecting a solution partitions the others into three categories:
Click on any polyline to select it. The dominance relationships update immediately.
Figure 3. 15 solutions with 4 objectives. Click a solution (shown in blue) to see which solutions dominate it (green), which it dominates (red), and which trade off with it (gray). Pareto-optimal solutions cannot be dominated by any other.
Adding objectives grows the front. Two objectives may leave only a small fraction non-dominated; ten objectives leave nearly everything Pareto-optimal because it's unlikely one solution beats another on every dimension. The curse of dimensionality applied to dominance.
Figure 4. 60 solutions. Add objectives to watch the Pareto front grow. With 2 objectives, few solutions are non-dominated. With 5, most are. The percentage counter tracks the growth.
Given the Pareto front, a decision-maker must choose one solution. A common approach is the weighted sum: assign a weight to each objective and optimize the total. Different weight vectors select different points on the Pareto front, sweeping out the trade-off surface.
As the weights change, you walk along the Pareto front, revealing how different priorities lead to different optimal designs.
Drag the weight sliders to change priorities, or click any polyline to find weights that favor it. The weighted-sum optimal design highlights in blue.
Figure 5. The 5-objective car design data with adjustable weights. Different weight combinations select different Pareto-optimal solutions. Click a polyline to reverse-engineer the weights that make it optimal. The weighted-sum approach lets a decision-maker explore the trade-off surface interactively.
Parallel coordinates are one of the few representations that show a high-dimensional Pareto front directly. Scatterplot matrices need panels and miss three-way interactions; radar charts fail beyond a handful of solutions. Each new objective just adds an axis.