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Nonsmooth Optimization Techniques on Riemannian Manifolds

Author

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  • S. Hosseini

    (Institute for Research in Fundamental Sciences (IPM))

  • M. R. Pouryayevali

    (Institute for Research in Fundamental Sciences (IPM))

Abstract

We present the notion of weakly metrically regular functions on manifolds. Then, a sufficient condition for a real valued function defined on a complete Riemannian manifold to be weakly metrically regular is obtained, and two optimization problems on Riemannian manifolds are considered. Moreover, we present a generalization of the Palais–Smale condition for lower semicontinuous functions defined on manifolds. Then, we use this notion to obtain necessary conditions of optimality for a general minimization problem on complete Riemannian manifolds.

Suggested Citation

  • S. Hosseini & M. R. Pouryayevali, 2013. "Nonsmooth Optimization Techniques on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 328-342, August.
    • Handle: RePEc:spr:joptap:v:158:y:2013:i:2:d:10.1007_s10957-012-0250-z
    DOI: 10.1007/s10957-012-0250-z
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    References listed on IDEAS

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    1. J. H. Wang & G. López & V. Martín-Márquez & C. Li, 2010. "Monotone and Accretive Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 691-708, September.
    2. A. Moldovan & L. Pellegrini, 2009. "On Regularity for Constrained Extremum Problems. Part 1: Sufficient Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 147-163, July.
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    Cited by:

    1. Fabiana R. de Oliveira & Orizon P. Ferreira, 2020. "Newton Method for Finding a Singularity of a Special Class of Locally Lipschitz Continuous Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 522-539, May.
    2. Orizon P. Ferreira & Célia Jean-Alexis & Alain Piétrus, 2017. "Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 624-651, December.
    3. Fabiana R. Oliveira & Fabrícia R. Oliveira, 2021. "A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 259-273, July.

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