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, Volume 22, Issue 3, pp 1148–1170 | Cite as

Concepts and techniques of optimization on the sphere

Original Paper

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.

Keywords

Sphere Convex function in the sphere Spheric constrained optimization Variational problem Monotone vector fields 

Mathematics Subject Classification

26B25 90C25 

Notes

Acknowledgments

The authors O. P. Ferreira was supported in part by FUNAPE/UFG, CNPq Grants 201112/2009-4, 475647/2006-8 and PRONEX–Optimization(FAPERJ/CNPq). A. N. Iusem was supported in part by CNPq grant no. 301280/86 and PRONEX-Otimização(FAPERJ/CNPq).

References

  1. Afsarii B, Tron R, René R (2013) On the convergence of gradient descent for finding the Riemannian center of mass. SIAM J Control Optim 51(3):2230–2260CrossRefGoogle Scholar
  2. Barani A, Pouryayevali MR (2009) Invariant monotone vector fields on Riemannian manifolds. Nonlin Anal 70(5):1850–1861CrossRefGoogle Scholar
  3. Dahl G, Leinaas JM, Myrheim J, Ovrum E (2007) A tensor product matrix approximation problem in quantum physics. Linear Algebra Appl 420(2–3):711–725CrossRefGoogle Scholar
  4. Das P, Chakraborti NR, Chaudhuri PK (2001) Spherical minimax location problem. Comput Optim Appl 18(3):311–326CrossRefGoogle Scholar
  5. Dennis JE, Jr, Schnabel RB (1996) Numerical methods for unconstrained optimization and nonlinear equations, vol 16 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), PhiladelphiaGoogle Scholar
  6. do Carmo MP (1976) Differential geometry of curves and surfaces. Prentice-Hall Inc, Englewood Cliffs (Translated from the Portuguese)Google Scholar
  7. do Carmo MP (1992) Riemannian geometry. Mathematics: theory and applications. Birkhäuser Boston Inc, Boston (Translated from the second Portuguese edition by Francis Flaherty)CrossRefGoogle Scholar
  8. Drezner Z, Wesolowsky GO (1983) Minimax and maximin facility location problems on a sphere. Naval Res Logist Quart 30(2):305–312CrossRefGoogle Scholar
  9. Ferreira OP, Iusem AN, Nemeth SZ (2013) Projections onto convex sets on the sphere. J Glob Optim 57:663–676CrossRefGoogle Scholar
  10. Ferreira OP, Oliveira P (1998) Subgradient algorithm on Riemannian manifolds. J Optim Theory Appl 97(1):93–104CrossRefGoogle Scholar
  11. Ferreira R, Xavier J, Costeira J, Barroso V (2008) Newton algorithms for riemannian distance related problems on connected locally symmetric manifolds. Thechnical Report: Institute for Systems and Robotics (ISR), Signal and Image Processing Group (SPIG), Instituto Superior Tecnico (IST)Google Scholar
  12. Fletcher PT, Venkatasubramanian S, Joshi S (2009) The geometric median on Riemannian manifolds with application to robust atlas estimation. NeuroImage 45:S143–S152CrossRefGoogle Scholar
  13. Han D, Dai HH, Qi L (2009) Conditions for strong ellipticity of anisotropic elastic materials. J Elast 97(1):1–13CrossRefGoogle Scholar
  14. He S, Li Z, Zhang S (2010) Approximation algorithms for homogeneous polynomial optimization with quadratic constraints. Math Program 125(2):353–383CrossRefGoogle Scholar
  15. Iqbal A, Ali S, Ahmad I (2012) On geodesic E-convex sets, geodesic E-convex functions and E-epigraphs. J Optim Theory Appl 155(1):239–251CrossRefGoogle Scholar
  16. Iusem A, Seeger A (2005) On pairs of vectors achieving the maximal angle of a convex cone. Math Program 104(2—-3):501–523CrossRefGoogle Scholar
  17. Iusem A, Seeger A (2009) Searching for critical angles in a convex cone. Math Program 120(1):3–25CrossRefGoogle Scholar
  18. Katz IN, Cooper L (1980) Optimal location on a sphere. Comput Math Appl 6(2):175–196CrossRefGoogle Scholar
  19. Laurent M (2009) Sums of squares, moment matrices and optimization over polynomials. In: Putinar M, Sullivant S (eds) Emerging applications of algebraic geometry, vol 149 of IMA Vol. Math. Appl. Springer, New York, pp 157–270Google Scholar
  20. Li C, Yao J-C (2012) Variational inequalities for set-valued vector fields on Riemannian manifolds: convexity of the solution set and the proximal point algorithm. SIAM J Control Optim 50(4):2486–2514CrossRefGoogle Scholar
  21. Li S-L, Li C, Liou Y-C, Yao J-C (2009) Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal 71(11):5695–5706CrossRefGoogle Scholar
  22. Ling C, Nie J, Qi L, Ye Y (2009) Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J Optim 20(3):1286–1310CrossRefGoogle Scholar
  23. Qi L (2005) Eigenvalues of a real supersymmetric tensor. J Symbolic Comput 40(6):1302–1324CrossRefGoogle Scholar
  24. Qi L, Teo KL (2003) Multivariate polynomial minimization and its application in signal processing. J Glob Optim 26(4):419–433CrossRefGoogle Scholar
  25. Qi L, Wang F, Wang Y (2009) \(Z\)-eigenvalue methods for a global polynomial optimization problem. Math Program 118(2):301–316CrossRefGoogle Scholar
  26. Reznick B (2000) Some concrete aspects of Hilbert’s 17th Problem. In: Real algebraic geometry and ordered structures (Baton Rouge, LA, 1996), vol 253 of Contemp. Math. Amer. Math. Soc., Providence, pp 251–272Google Scholar
  27. Sakai T (1996) Riemannian geometry, volume 149 of Translations of Mathematical Monographs. American Mathematical Society, Providence (Translated from the 1992 Japanese original by the author)Google Scholar
  28. Smith ST (1994) Optimization techniques on Riemannian manifolds. In: Hamiltonian and gradient flows, algorithms and control, vol 3 of Fields Inst. Commun. Amer. Math. Soc., Providence, pp 113–136Google Scholar
  29. So AMC (2011) Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems. Math Program 129(2):357–382CrossRefGoogle Scholar
  30. Weiland S, van Belzen F (2010) Singular value decompositions and low rank approximations of tensors. IEEE Trans Signal Process 58(3):1171–1182CrossRefGoogle Scholar
  31. Xue G-L (1994) A globally convergent algorithm for facility location on a sphere. Comput Math Appl 27(6):37–50CrossRefGoogle Scholar
  32. Xue GL (1995) On an open problem in spherical facility location. Numer Algorithms 9(1–2):1–12CrossRefGoogle Scholar
  33. Zhang L (2003) On the convergence of a modified algorithm for the spherical facility location problem. Oper Res Lett 31(2):161–166CrossRefGoogle Scholar
  34. Zhang X, Ling C, Qi L (2012) The best rank-1 approximation of a symmetric tensor and related spherical optimization problems. SIAM J Matrix Anal Appl 33(3):806–821CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2014

Authors and Affiliations

  1. 1.IME/UFGGoiâniaBrazil
  2. 2.Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil
  3. 3.School of Mathematics, The University of BirminghamBirminghamUK

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