Multi-objective variational curves

TL;DR

This paper formulates the problem of finding Riemannian cubics as a multi-objective optimization, constructs Pareto fronts on spheres and tori, and reveals novel disconnected fronts with multiple solutions.

Contribution

It introduces a multi-objective perspective to Riemannian cubics and uncovers the first known disconnected Pareto front with multiple solutions on a torus.

Findings

Pareto fronts constructed for spheres and tori

Discovered disconnected Pareto front on the torus

Identified multiple Riemannian cubics with same boundary data

Abstract

Riemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared norms of the velocity and acceleration of a curve on a Riemannian manifold. We view this variational problem of finding a curve as a multi-objective optimization problem and construct the Pareto fronts for some given instances where the manifold is a sphere and where the manifold is a torus. The Pareto front for the curves on the torus turns out to be particularly interesting: the front is disconnected and it reveals two distinct Riemannian cubics with the same boundary data, which is the first known nontrivial instance of this kind. We also discuss some convexity conditions involving the Pareto fronts for curves on general Riemannian manifolds.