Multi-objective Variational Curves

C. Yalçın Kaya (), Lyle Noakes () and Erchuan Zhang ()
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C. Yalçın Kaya: University of South Australia
Lyle Noakes: University of Western Australia
Erchuan Zhang: University of Western Australia

Journal of Optimization Theory and Applications, 2024, vol. 201, issue 2, No 17, 955-975

Abstract: Abstract Riemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared $$L^2$$ L 2 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.

Keywords: Riemannian cubics in tension; Multi-objective optimal control; Pareto front; Numerical methods; 58E17; 49M25; 53Z99 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02427-0

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