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The Difference of Convex Algorithm on Hadamard Manifolds

Author

Listed:
  • Ronny Bergmann

    (Norwegian University of Science and Technology)

  • Orizon P. Ferreira

    (Federal University of Goiás)

  • Elianderson M. Santos

    (Federal Institute of Education, Science and Technology of Maranhão)

  • João Carlos O. Souza

    (Federal University of Piauí)

Abstract

In this paper, we propose a Riemannian version of the difference of convex algorithm (DCA) to solve a minimization problem involving the difference of convex (DC) function. The equivalence between the classical and simplified Riemannian versions of the DCA is established. We also prove that under mild assumptions the Riemannian version of the DCA is well defined and every cluster point of the sequence generated by the proposed method, if any, is a critical point of the objective DC function. Some duality relations between the DC problem and its dual are also established. To illustrate the algorithm’s effectiveness, some numerical experiments are presented.

Suggested Citation

  • Ronny Bergmann & Orizon P. Ferreira & Elianderson M. Santos & João Carlos O. Souza, 2024. "The Difference of Convex Algorithm on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 201(1), pages 221-251, April.
    • Handle: RePEc:spr:joptap:v:201:y:2024:i:1:d:10.1007_s10957-024-02392-8
    DOI: 10.1007/s10957-024-02392-8
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    References listed on IDEAS

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    1. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    2. J. Souza & P. Oliveira, 2015. "A proximal point algorithm for DC fuctions on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 63(4), pages 797-810, December.
    3. NESTEROV , Yu. & TODD, Mike, 2002. "On the Riemannian geometry defined by self-concordant barriers and interior-point methods," LIDAM Reprints CORE 1595, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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