Existence of solutions for variational inequalities on Riemannian manifolds. (English) Zbl 1180.58012
This paper deals with a class of variational inequalities on Riemannian manifolds. By means of variational techniques, the authors establish the existence of a unique solution. Connections with the related constrained optimization problem are also given in the present paper.
Reviewer: Teodora-Liliana Rădulescu (Craiova)
MSC:
| 58E35 | Variational inequalities (global problems) in infinite-dimensional spaces |
| 49Q10 | Optimization of shapes other than minimal surfaces |
References:
| [1] | Harker, P. T.; Pang, J. S., Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Program., 48, 161-220 (1990) · Zbl 0734.90098 |
| [2] | Patriksson, M., Nonlinear Programming and Variational Inequality Problems: A Unified Approach (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0913.65058 |
| [3] | Adler, R.; Dedieu, J. P.; Margulies, J.; Martens, M.; Shub, M., Newton method on Riemannian manifolds and a geometric model for human spine, IMA J. Numer. Anal., 22, 1-32 (2002) |
| [4] | Edelman, A.; Arias, T. A.; Smith, T., The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20, 303-353 (1998) · Zbl 0928.65050 |
| [5] | Gabay, D., Minimizing a differentiable function over a differentiable manifold, J. Optim. Theory Appl., 37, 177-219 (1982) · Zbl 0458.90060 |
| [6] | Németh, S. Z., Variational inequalities on Hadamard manifolds, Nonlinear Anal., 52, 1491-1498 (2003) · Zbl 1016.49012 |
| [7] | Smith, S., Optimization techniques on Riemannian manifolds, (Bloch, Anthony, Hamiltonian and Gradient Flows, Algorithms and Control. Hamiltonian and Gradient Flows, Algorithms and Control, Fields Institute Communications, vol. 3 (1994), American Mathematical Society: American Mathematical Society Providence, RI), 113-136 · Zbl 0816.49032 |
| [8] | S.T. Smith, Geometric Optimization Methods for Adaptive Filtering, Ph.D. Thesis, Division of Applied Science, Harvard University, Cambridge, MA, 1993; S.T. Smith, Geometric Optimization Methods for Adaptive Filtering, Ph.D. Thesis, Division of Applied Science, Harvard University, Cambridge, MA, 1993 |
| [9] | Udrişte, C., Convex Functions and Optimization Methods on Riemannian Manifolds (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0932.53003 |
| [10] | Alvarez, F.; Bolte, J.; Munier, J., A unifying local convergence result for Newton’s method in Riemannian manifolds, Found. Comput. Math., 8, 197-226 (2008) · Zbl 1147.58008 |
| [11] | Ferreira, O. P.; Svaiter, B. F., Kantorovich’s theorem on Newton’s method in Riemannian manifolds, J. Complexity, 18, 304-329 (2002) · Zbl 1003.65057 |
| [12] | Dedieu, J. P.; Priouret, P.; Malajovich, G., Newton’s method on Riemannian manifolds: Covariant alpha theory, IMA J. Numer. Anal., 23, 395-419 (2003) · Zbl 1047.65037 |
| [13] | Li, C.; Wang, J. H., Newton’s method on Riemannian manifolds: Smale’s point estimate theory under the \(\gamma \)-condition, IMA J. Numer. Anal., 26, 228-251 (2006) · Zbl 1094.65052 |
| [14] | Ferreira, O. P.; Oliveira, P. R., Proximal point algorithm on Riemannian manifolds, Optimization, 51, 257-270 (2002) · Zbl 1013.49024 |
| [15] | Li, C.; López, G.; Martín-Máquez, V., Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. London Math. Soc. (2009) · Zbl 1171.58001 |
| [16] | Boothby, W. M., An Introduction to Differentiable Manifolds and Riemannian Geometry (1986), Academic Press: Academic Press New York · Zbl 0596.53001 |
| [17] | DoCarmo, M. P., Riemannian Geometry (1992), Birkhauser: Birkhauser Boston · Zbl 0752.53001 |
| [18] | Petersen, P., Riemannian Geometry, GTM 171 (2006), Springer · Zbl 1220.53002 |
| [19] | Sakai, T., (Riemannian Geometry. Riemannian Geometry, Translations of Mathematical Monographs, vol. 149 (1992), American Mathematical Society) |
| [20] | Walter, R., On the metric projections onto convex sets in Riemannian spaces, Arch. Math., XXV, 91-98 (1974) · Zbl 0311.53054 |
| [21] | Cheeger, J.; Gromoll, D., On the structure of complete manifolds of nonnegative curvature, Ann. of Math., 96, 413-443 (1972) · Zbl 0246.53049 |
| [22] | D. Gromoll, W. Klingenberg, W. Meyer, Riemannsche Geometri im GroBen, in: Lect. Notes in Math. vol. 55, Berlin, Heidelberg, New York, 1968; D. Gromoll, W. Klingenberg, W. Meyer, Riemannsche Geometri im GroBen, in: Lect. Notes in Math. vol. 55, Berlin, Heidelberg, New York, 1968 · Zbl 0155.30701 |
| [23] | Németh, S. Z., Five kinds of monotone vector fields, Pure Appl. Math., 9, 417-428 (1999) · Zbl 0937.58010 |
| [24] | Németh, S. Z., Monotone vector fields, Publ. Math., 54, 437-449 (1999) · Zbl 0963.47036 |
| [25] | Németh, S. Z., Geodesic monotone vector fields, Lobachevskii J. Math., 5, 13-28 (1999) · Zbl 0970.53021 |
| [26] | Németh, S. Z., Homeomorphisms and monotone vector fields, Publ. Math., 58, 4, 707-716 (2001) · Zbl 0980.53051 |
| [27] | Da Cruz Neto, J. X.; Ferreira, O. P.; Lucambio Pérez, L. R., Contributions to the study of monotone vector fields, Acta Math. Hungar., 94, 4, 307-320 (2002) · Zbl 0997.53012 |
| [28] | Da Cruz Neto, J. X.; Ferreira, O. P.; Lucambio Pérez, L. R.; Németh, S. Z., Convex- and monotone-transformable mathematical programming problems and a proximal-like point method, J. Global Optim., 35, 1, 53-69 (2006) · Zbl 1104.90035 |
| [29] | Da Cruz Neto, J. X.; Ferreira, O. P.; Lucambio Pérez, L. R., Monotone point-to-set vector fields, Dedicated to Professor Constantin Udrişte, Balkan J. Geom. Appl., 5, 1, 69-79 (2000) · Zbl 0978.90076 |
| [30] | Ferreira, O. P.; Lucambio Pérez, L. R.; Németh, S. Z., Singularities of monotone vector fields and an extragradient-type algorithm, J. Global Optim., 31, 133-151 (2005) · Zbl 1229.58007 |
| [31] | Isac, G.; Németh, S. Z., (Scalar and Asymptotic Scalar Derivatives, Theory and Applications. Scalar and Asymptotic Scalar Derivatives, Theory and Applications, Springer Optimization and its Applications, vol. 13 (2008), Springer: Springer New York) · Zbl 1159.46002 |
| [32] | Karcher, H., Schnittort und konvexe Mengen in vollständigen Riemannschen Mannigfaltigkeiten, Math. Ann., 177, 105-121 (1968) · Zbl 0157.28801 |
| [33] | Németh, S. Z., Monotonicity of the complementary vector field of a non-expansive map, Acta Math. Hungar., 84, 189-197 (1999) · Zbl 0992.53016 |
| [34] | Rapcsák, T., Smooth Nonlinear Optimization in \(R^n (1997)\), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 1009.90109 |
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