Constructing Pareto Sets Using Noether's Second Theorem

Abstract

This paper presents an analytical approach to constructing the Pareto set in multiobjective variational control problems, based on Noether's second theorem. A fundamental connection is established between the gauge symmetries of the dynamical system describing the plant and the structure of the Pareto set. It is shown that the Pareto front can be interpreted as a conservation law arising from the invariance of the system and the quality criteria with respect to a symmetry group. A gauge regularization concept is proposed, which allows eliminating variables that do not affect the criteria without changing the Pareto set, thereby reducing the dimension of the solution space. Numerical examples are provided for the linear-quadratic regulator with two conflicting criteria and for the standard ZDT1 test problem.

13 pages