A Propagation Property of the Pareto Order with Applications to Multiobjective Optimization

TMLR Paper6229 Authors

Abstract: We establish a propagation property of the strict Pareto dominance order $\prec$ on $\mathbb{R}^k$ and present its application for efficient nondominated sorting and Pareto archive maintenance in multiobjective optimization. Precisely, if $u,v\in\mathbb{R}^k$ are mutually nondominated ($(u\nprec v) \wedge\ (v\nprec u)$), then no $q\in\mathbb{R}^k$ can satisfy $u\prec q\prec v$ or $v\prec q\prec u$. We give algebraic and geometric proofs, the latter via containment of strict down-sets (lower orthants). As a corollary, we state and prove a \emph{post-witness $u \prec q$– elimination rule} for Pareto archive $S$ insertion: once a witness $w\in S$ with $q\prec w$ is found, no remaining $u \in S \setminus\{w\}$ can dominate $q$. We therefore can skip all subsequent ``$u\prec q$?'' checks and only test ``$q\prec u$?'' to remove dominated vectors. We provide pseudocode for the resulting archive-insertion routine, and outline extensions to weak, $\varepsilon$-, and noisy dominance. Finally, under a standard random-input model in which points are drawn independently from a continuous distribution on $\mathbb{R}^k$ (general position almost surely), we derive a closed-form expression for the expected post-witness fraction of the remaining “$u\prec q$?” comparisons (over $u\in S\setminus\{w\}$) that become unnecessary once a first witness $w$ with $q\prec w$ is identified. The formula reveals how savings in comparison-checks scale with dimension $k$ (and archive size), justifies witness-first heuristic scanning orders, and provides a reproducible baseline for empirical evaluation of dominance comparison-count reductions in archive insertion and nondominated sorting implementations. This probabilistic baseline complements the deterministic post-witness exclusion guaranteed by the propagation property in mutually nondominated curated archives.

Submission Length: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Tian_Li1

Submission Number: 6229