Singularities of Monotone Vector Fields and an Extragradient-type Algorithm
- 203 Downloads
Bearing in mind the notion of monotone vector field on Riemannian manifolds, see [12--16], we study the set of their singularities and for a particularclass of manifolds develop an extragradient-type algorithm convergent to singularities of such vector fields. In particular, our method can be used forsolving nonlinear constrained optimization problems in Euclidean space, with a convex objective function and the constraint set a constant curvature Hadamard manifold. Our paper shows how tools of convex analysis on Riemannian manifolds can be used to solve some nonconvex constrained problem in a Euclidean space.
Keywordsextragradient algorithm global optimization Hadamard manifold monotone vector field
Unable to display preview. Download preview PDF.
- Burachik R., Sagastiz l C., Svaiter B.F. (1999). Bundle methods for maximal monotone operators, Ill-posed variational problems and regularization techniques (Trier, 1998), In: Lecture Notes in Econom. and Math. Systems, 477: Springer, Berlin, pp. 49-64.Google Scholar
- da Cruz Neto J.X., Ferreira O.P. and Lucambio Pérez L.R. (2002). Contribution to the study of monotone vectorfields, to be pubished in Acta Mathematica Hungarica.Google Scholar
- Németh, S.Z. 1999Five kinds of monotone vectorfieldsPU.M.A.9417428Google Scholar
- Németh S.Z. (2003). Variational inequalities on Hadamard manifolds, to be published in Nonlinear Analysis: Theory, Methods & Applications, 52: 1491-1498Google Scholar
- Sakai, T. 1996Riemannian Geometry, Translations of Mathematical Monographs 149American Mathematical SocietyProvidence R.IGoogle Scholar
- Shiga, K. (1984). Hadamard Manifolds, Advanced Studies in Pure Mathematics 3, Geometry of Geodesics and Related Topic, 239-281.Google Scholar
- Smith S.T. (1994). Optimization techniques on Riemannian Manifolds, Fields Institute Communications 3, American Mathematical Society, Providence R.I., 113-146Google Scholar
- Udri¸ste, C. (1994). Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and its Applications 297, KluwerAcademic Publishers.Google Scholar
- Zeidler, E. (1990). Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators, Springer Verlag.Google Scholar