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Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off

Author

Listed:
  • J. X. Cruz Neto

    (Universidade Federal do Piauí)

  • P. R. Oliveira

    (Universidade Federal do Rio de Janeiro)

  • P. A. Soares

    (Universidade Etadual do Piauí)

  • A. Soubeyran

    (Aix-Marseille University, CNRS & EHESS)

Abstract

In this paper, we consider minimization problems with constraints. We show that, if the set of constraints is a Finslerian manifold of non-positive flag curvature, and the objective function is differentiable and satisfies the Kurdyka-Lojasiewicz property, then the proximal point method can be naturally extended to solve this class of problems. We prove that the sequence generated by our method is well defined and converges to a critical point. We show how tools of Finslerian geometry, specifically non-symmetrical metrics, can be used to solve non-convex constrained problems in Euclidean spaces. As an application, we give one result regarding decision-making speed and costs related to change.

Suggested Citation

  • J. X. Cruz Neto & P. R. Oliveira & P. A. Soares & A. Soubeyran, 2014. "Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 873-891, September.
    • Handle: RePEc:spr:joptap:v:162:y:2014:i:3:d:10.1007_s10957-013-0483-5
    DOI: 10.1007/s10957-013-0483-5
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    1. Herbert A. Simon, 1955. "A Behavioral Model of Rational Choice," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 69(1), pages 99-118.
    2. James G. March, 1991. "Exploration and Exploitation in Organizational Learning," Organization Science, INFORMS, vol. 2(1), pages 71-87, February.
    3. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
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    Cited by:

    1. G. C. Bento & J. X. Cruz Neto & P. A. Soares & A. Soubeyran, 2022. "A new regularization of equilibrium problems on Hadamard manifolds: applications to theories of desires," Annals of Operations Research, Springer, vol. 316(2), pages 1301-1318, September.

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