Response Optimization

Pareto Frontier (2D, Non-Dominance Filter)

Filters a set of points down to the non-dominated solutions — the Pareto frontier of a multi-objective optimisation.

Statusvalidated

Version1.0.0

Description

When goals compete (e.g. maximise mean AND minimise variance), there is rarely a single solution that optimises every goal at once. The Pareto frontier is the set of points where no improvement in one goal is possible without degrading another. Algorithm: a point a dominates b if a is ≤ b in every objective and strictly smaller in at least one. Non-dominated points (those not dominated by any other) make up the frontier. The function expects minimisation — flip the sign for maximisation goals.

Formulas

Dominanz

a ≻ b ⟺ (\forall i : a_{i} \leq b_{i}) \land (\exists i : a_{i} < b_{i})

a dominates b iff a is ≤ b in every component and strictly smaller in at least one

Frontier

P = {x \in X : ∄ y \in X mit y ≻ x}

Pareto frontier: points not dominated by any other

Assumptions

All objectives are minimised (flip the sign for maximisation)
All points share the same dimension (number of objectives)

Limitations

Naïve O(n²) algorithm — consider more efficient methods for n > 10 000 points
Beyond 3 objectives the Pareto front quickly grows large and hard to interpret in practice

References

Pareto, V. (1906). Manuale di Economia Politica
Deb, K. (2001). Multi-Objective Optimization Using Evolutionary Algorithms, Wiley — Chapter 2.4
Myers, R.H., Montgomery, D.C., Anderson-Cook, C.M. (2016). Response Surface Methodology, 4th Ed., Wiley — Chapter 6.5 (Multi-Response Optimization)