epsilondominance

epsilondominance is a relaxation of Pareto dominance used in multiobjective optimization. It introduces a tolerance parameter, epsilon, to account for imprecision, noise, or the desire for a more compact frontier. In a minimization setting with objective vector f(x) = (f1(x), ..., fm(x)) and a nonnegative tolerance ε (which may be a scalar or a vector ε ∈ R_+^m), solution x ε-dominates y if, for every objective i, fi(x) ≤ fi(y) + ε_i, and there exists at least one objective k for which fk(x) ≤ fk(y) − ε_k (in the scalar case, there is a single ε and the last condition reads fk(x) ≤ fk(y) − ε). Intuitively, x is no worse than y across all objectives within a margin ε, and strictly better in at least one objective by more than ε. When ε = 0 (or ε_i = 0 for all i), epsilondominance reduces to standard Pareto dominance.

epsilondominance is often used to discretize the objective space into epsilon-sized boxes, effectively clustering nearby solutions and reducing duplicates on the Pareto front. This leads to archive or selection mechanisms that retain a representative, nonredundant set of solutions, improving stability and diversity, especially in the presence of noisy or expensive evaluations. It also helps manage the size of the nondominated set in many-objective problems and facilitates comparisons across generations in evolutionary multiobjective algorithms.

The choice of ε (scalar or per-objective) reflects the desired granularity and problem scale. Smaller ε yields a finer front closer to true Pareto dominance, while larger ε promotes a more compact, robust set of solutions and can smooth out minor variations in objective values.