Advertisement

On the spherical convexity of quadratic functions

  • O. P. Ferreira
  • S. Z. Németh
Article
  • 40 Downloads

Abstract

In this paper we study the spherical convexity of quadratic functions on spherically convex sets. In particular, conditions characterizing the spherical convexity of quadratic functions on spherical convex sets associated to the positive orthants and Lorentz cone are given.

Keywords

Spheric convexity Quadratic functions Positive orthant Lorentz cone 

Notes

Acknowledgements

The authors are grateful to Michal Kočvara and Kay Magaard for many helpful conversations.

References

  1. 1.
    Dahl, G., Leinaas, J.M., Myrheim, J., Ovrum, E.: A tensor product matrix approximation problem in quantum physics. Linear Algebr. Appl. 420, 711–725 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Das, P., Chakraborti, N.R., Chaudhuri, P.K.: Spherical minimax location problem. Comput. Optim. Appl. 18, 311–326 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dennis Jr., J .E., Schnabel, R .B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Classics in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1996)CrossRefGoogle Scholar
  4. 4.
    Drezner, Z., Wesolowsky, G.O.: Minimax and maximin facility location problems on a sphere. Naval Res. Logist. Quart. 30, 305–312 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ferreira, O.P., Iusem, A.N., Németh, S.Z.: Projections onto convex sets on the sphere. J. Global Optim. 57, 663–676 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ferreira, O.P., Iusem, A.N., Németh, S.Z.: Concepts and techniques of optimization on the sphere. TOP 22, 1148–1170 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fletcher, P.T., Venkatasubramanian, S., Joshi, S.: The geometric median on riemannian manifolds with application to robust atlas estimation. NeuroImage 45, S143–S152 (2009)CrossRefGoogle Scholar
  8. 8.
    Han, D., Dai, H.H., Qi, L.: Conditions for strong ellipticity of anisotropic elastic materials. J. Elast. 97, 1–13 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Isac, G., Németh, S .Z.: Scalar derivatives and scalar asymptotic derivatives: properties and some applications. J. Math. Anal. Appl. 278(1), 149–170 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Isac, G., Németh, S.Z.: Scalar derivatives and scalar asymptotic derivatives. An Altman type fixed point theorem on convex cones and some applications. J. Math. Anal. Appl. 290, 452–468 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Isac, G., Németh, S.Z.: Duality in multivalued complementarity theory by using inversions and scalar derivatives. J. Global Optim. 33, 197–213 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Isac, G., Németh, S.Z.: Duality in nonlinear complementarity theory by using inversions and scalar derivatives. Math. Inequal. Appl. 9, 781–795 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Isac, G., Németh, S.Z.: Duality of implicit complementarity problems by using inversions and scalar derivatives. J. Optim. Theory Appl. 128, 621–633 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Isac, G., Németh,.S .Z.: Scalar and Asymptotic Scalar Derivatives. Theory and Applications. Springer Optimization and Its Applications, vol. 13. Springer, New York (2008)zbMATHGoogle Scholar
  15. 15.
    Karamardian, S., Schaible, S.: Seven kinds of monotone maps. J. Optim. Theory Appl. 66, 37–46 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and Its Applications, vol. 149, pp. 157–270. Springer, New York (2009)CrossRefGoogle Scholar
  17. 17.
    Ling, C., Nie, J., Qi, L., Ye, Y.: Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J. Optim. 20, 1286–1310 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Németh, S .Z.: A scalar derivative for vector functions. Riv. Mat. Pura Appl. 10, 7–24 (1992)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Németh, S.Z.: Scalar derivatives and spectral theory. Mathematica (Cluj) 35(58), 49–57 (1993)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Németh, S.Z.: Scalar derivatives and conformity. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 40(1997), 99–105 (1998)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Németh, S.Z.: Five kinds of monotone vector fields. Pure Math. Appl. 9, 417–428 (1998)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Németh, S.Z.: Scalar derivatives in Hilbert spaces. Positivity 10, 299–314 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Qi, L., Teo, K.L.: Multivariate polynomial minimization and its application in signal processing. J. Global Optim. 26, 419–433 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Qi, L., Wang, F., Wang, Y.: \(Z\)-eigenvalue methods for a global polynomial optimization problem. Math. Program. 118, 301–316 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Reznick, B.: Some concrete aspects of Hilbert’s 17th problem. In: Delzell, C.N., Madden, J.J. (eds.) Real Algebraic Geometry and Ordered Structures (Baton Rouge, LA, 1996). Contemporary Mathematics, vol. 253, pp. 251–272. American Mathematical Society, Providence (2000)CrossRefGoogle Scholar
  27. 27.
    Smith, S .T.: Optimization techniques on Riemannian manifolds. In: Bloch, A.M. (ed.) Hamiltonian and Gradient Flows, Algorithms and Control. Fields Institute Communications, vol. 3, pp. 113–136. Americam Mathematical Society, Providence (1994)Google Scholar
  28. 28.
    So, A.M.-C.: Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems. Math. Program. 129, 357–382 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Weiland, S., van Belzen, F.: Singular value decompositions and low rank approximations of tensors. IEEE Trans. Signal Process. 58, 1171–1182 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Xue, G.-L.: A globally convergent algorithm for facility location on a sphere. Comput. Math. Appl. 27, 37–50 (1994)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Xue, G.L.: On an open problem in spherical facility location. Numer. Algorithms 9, 1–12 (1995)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Zhang, L.: On the convergence of a modified algorithm for the spherical facility location problem. Oper. Res. Lett. 31, 161–166 (2003)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhang, X., Ling, C., Qi, L.: The best rank-1 approximation of a symmetric tensor and related spherical optimization problems. SIAM J. Matrix Anal. Appl. 33, 806–821 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IME/UFG, Avenida Esperança, s/n, Campus SamambaiaGoiâniaBrazil
  2. 2.School of MathematicsUniversity of BirminghamEdgbastonUK

Personalised recommendations