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Multi-objective Variational Curves

Author

Listed:
  • C. Yalçın Kaya

    (University of South Australia)

  • Lyle Noakes

    (University of Western Australia)

  • Erchuan Zhang

    (University of Western Australia
    Sun Yat-sen University)

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.

Suggested Citation

  • C. Yalçın Kaya & Lyle Noakes & Erchuan Zhang, 2024. "Multi-objective Variational Curves," Journal of Optimization Theory and Applications, Springer, vol. 201(2), pages 955-975, May.
    • Handle: RePEc:spr:joptap:v:201:y:2024:i:2:d:10.1007_s10957-024-02427-0
    DOI: 10.1007/s10957-024-02427-0
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    References listed on IDEAS

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    1. C. Kaya & Helmut Maurer, 2014. "A numerical method for nonconvex multi-objective optimal control problems," Computational Optimization and Applications, Springer, vol. 57(3), pages 685-702, April.
    2. Henri Bonnel & C. Yalçın Kaya, 2010. "Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 93-112, October.
    3. C. Yalçın Kaya & Helmut Maurer, 2023. "Optimization over the Pareto front of nonconvex multi-objective optimal control problems," Computational Optimization and Applications, Springer, vol. 86(3), pages 1247-1274, December.
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