Noether's Second Theorem and the Pareto Front Structure in Multi-Criteria Variational Problems
Abstract
This paper establishes a rigorous connection between Noether's second theorem and the structure of Pareto fronts in multi-criteria variational problems. We prove that for a family of functionals invariant under a common gauge Lie group, the regular part of the Pareto front is contained in the union of solutions to weighted Euler--Lagrange equations, and that on Pareto-optimal sections, combinations of Noether currents are conserved. Gauge symmetries are shown to impose additional constraints on the Lagrange multipliers via Bianchi-type identities, reducing the dimension of the admissible weight space. Two practical approaches are derived from this result: dimensionality reduction via gauge fixing and scalarization with penalty terms. The theory is validated on three classical problems: the linear-quadratic regulator with two criteria, a model predictive control problem with a minimax criterion, and the double integrator with control constraints. In each case, the Pareto front is shown to satisfy an exact conservation law, and the analytical structure of the front is recovered from the symmetries of the problem. The results demonstrate that gauge functions (null Lagrangians) can restore invariants even when the dynamics break the symmetry, and that control constraints do not destroy the underlying conservation laws but merely restrict the admissible set. The proposed approach yields a Pareto front with minimal deviation from the theoretical curve, outperforming known multi-objective algorithms, and provides a new foundation for the analytical treatment of multi-criteria optimization in nonlinear dynamics and control.
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BibTeX
@misc{nikulchev2026noethers,
title = {{Noether's Second Theorem and the Pareto Front Structure in Multi-Criteria Variational Problems}},
author = {Evgeny Nikulchev},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06920},
eprint = {2611.06920},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06920},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Noether's Second Theorem and the Pareto Front Structure in Multi-Criteria Variational Problems
AU - Evgeny Nikulchev
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.06920
UR - https://arxiv.org/abs/2611.06920
AB - This paper establishes a rigorous connection between Noether's second theorem and the structure of Pareto fronts in multi-criteria variational problems. We prove that for a family of functionals invariant under a common gauge Lie group, the regular part of the Pareto front is contained in the union of solutions to weighted Euler--Lagrange equations, and that on Pareto-optimal sections, combinations of Noether currents are conserved. Gauge symmetries are shown to impose additional constraints on the Lagrange multipliers via Bianchi-type identities, reducing the dimension of the admissible weight space. Two practical approaches are derived from this result: dimensionality reduction via gauge fixing and scalarization with penalty terms. The theory is validated on three classical problems: the linear-quadratic regulator with two criteria, a model predictive control problem with a minimax criterion, and the double integrator with control constraints. In each case, the Pareto front is shown to satisfy an exact conservation law, and the analytical structure of the front is recovered from the symmetries of the problem. The results demonstrate that gauge functions (null Lagrangians) can restore invariants even when the dynamics break the symmetry, and that control constraints do not destroy the underlying conservation laws but merely restrict the admissible set. The proposed approach yields a Pareto front with minimal deviation from the theoretical curve, outperforming known multi-objective algorithms, and provides a new foundation for the analytical treatment of multi-criteria optimization in nonlinear dynamics and control.
ER -