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Concepts and techniques of optimization on the sphere

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  • O. Ferreira
  • A. Iusem
  • S. Németh

Abstract

In this paper some concepts and techniques of Mathematical Programming are extended in an intrinsic way from the Euclidean space to the sphere. In particular, the notion of convex functions, variational problem and monotone vector fields are extended to the sphere and several characterizations of these notions are shown. As an application of the convexity concept, necessary and sufficient optimality conditions for constrained convex optimization problems on the sphere are derived. Copyright Sociedad de Estadística e Investigación Operativa 2014

Suggested Citation

  • O. Ferreira & A. Iusem & S. Németh, 2014. "Concepts and techniques of optimization on the sphere," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 1148-1170, October.
    • Handle: RePEc:spr:topjnl:v:22:y:2014:i:3:p:1148-1170
    DOI: 10.1007/s11750-014-0322-3
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    References listed on IDEAS

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    1. O. Ferreira & A. Iusem & S. Németh, 2013. "Projections onto convex sets on the sphere," Journal of Global Optimization, Springer, vol. 57(3), pages 663-676, November.
    2. Akhlad Iqbal & Shahid Ali & I. Ahmad, 2012. "On Geodesic E-Convex Sets, Geodesic E-Convex Functions and E-Epigraphs," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 239-251, October.
    3. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
    4. O. P. Ferreira & P. R. Oliveira, 1998. "Subgradient Algorithm on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 93-104, April.
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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. Henri Bonnel & Léonard Todjihoundé & Constantin Udrişte, 2015. "Semivectorial Bilevel Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 464-486, November.
    3. Peng Zhang & Gejun Bao, 2018. "An Incremental Subgradient Method on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 711-727, March.
    4. Orizon Pereira Ferreira & Sándor Zoltán Németh & Lianghai Xiao, 2020. "On the Spherical Quasi-convexity of Quadratic Functions on Spherically Subdual Convex Sets," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 1-21, October.
    5. Teles A. Fernandes & Orizon P. Ferreira & Jinyun Yuan, 2017. "On the Superlinear Convergence of Newton’s Method on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 828-843, June.
    6. O. P. Ferreira & S. Z. Németh, 2019. "On the spherical convexity of quadratic functions," Journal of Global Optimization, Springer, vol. 73(3), pages 537-545, March.

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