Concepts and techniques of optimization on the sphere
- 427 Downloads
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.
KeywordsSphere Convex function in the sphere Spheric constrained optimization Variational problem Monotone vector fields
Mathematics Subject Classification26B25 90C25
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).
- 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
- do Carmo MP (1976) Differential geometry of curves and surfaces. Prentice-Hall Inc, Englewood Cliffs (Translated from the Portuguese)Google Scholar
- 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
- 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
- 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
- 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
- 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