Abstract
In this paper, we investigate the proximal point algorithm (in short PPA) for variational inequalities with pseudomonotone vector fields on Hadamard manifolds. Under weaker assumptions than monotonicity, we show that the sequence generated by PPA is well defined and prove that the sequence converges to a solution of variational inequality, whenever it exists. The results presented in this paper generalize and improve some corresponding known results given in literatures.
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Barani A., Pouryayevali M.R.: Invex sets and preinvex functions on Riemannian manifolds. J. Math. Anal. Appl. 328, 767–779 (2007)
Barani A., Pouryayevali M.R.: Invariant monotone vector fields on Riemannian manifolds. Nonlinear Anal. 70, 1850–1861 (2009)
Bento G.C., Ferreira O.P., Oliveira P.R.: Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds. Nonlinear Anal. 73, 564–572 (2010)
Cruz Neto, J.X.D., Ferreira ,O.P., Lucambio Pérez, L.R.: Monotone point-to-set vector fields. Balkan J. Geom. Appl. 5, 69–79 (2000) (Dedicated to Professor Constantin Udriste)
Cruz Neto J.X.D., Ferreira O.P., Pérez L.R.L.: Contributions to the study of monotone vector fields. Acta Math. Hung. 94, 307–320 (2002)
Cruz Neto J.X.D., Ferreira O.P., Pérez L.R.L., Németh S.Z.: Convex- and monotone-transformable methematical programming problems and a proximal-like point method. J. Glob. Optim. 35, 53–69 (2006)
Daniilidis A., Hadjisavvas N.: Coercivity conditions and variational inequalities. Math. Program. 86, 433–438 (1999)
Facchinei F., Pang J.S.: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, New York (2003)
Farouq N.E.: Pseudomonotone variational inequalities: convergence of proximal methods. J. Optim. Theory Appl. 109, 311–326 (2001)
Ferreira O.P., Oliveira P.R.: Subgradient algorithm on Riemannian manifolds. J. Optim. Theory Appl. 97, 93–104 (1998)
Ferreira O.P., Oliveira P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002)
Ferreira O.P., Pérez L.R.L., Németh S.Z.: Singularities of monotone vector fields and an extragradient-type algorithm. J. Glob. Optim. 31, 133–151 (2005)
Fukushima M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992)
Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems and Variational Models. Kluwer, Boston (2001)
Hu Y.H., Song W.: Weak sharp solutions for variational inequalities in Banach spaces. J. Math. Anal. Appl. 374, 118–132 (2011)
Isac G., Németh S.Z.: Scalar and Asymptotic Scalar Derivatives Theory and Applications. Springer, New York (2008)
Kaplan A., Tichatschke R.: Proximal point methods and nonconvex optimization. J. Glob. Optim. 13, 389–406 (1998)
Li C., López G., Martín-Márquez V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663–683 (2009)
Li C., López G., Martín-Márquez V.: Iterative algorithms for nonexpansive mapping in Hadamard manifolds. Taiwan. J. Math. 14, 541–559 (2010)
Li, C., López, G., Martín-Márquez, V., Wang, J.H.: Resolvents of set-valued monotone vector fields in Hadamard manifolds. Set Valued Var. Anal. doi:10.1007/s11228-010-0169-1
Li S.L., Li C., Liou Y.C., Yao J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71, 5695–5706 (2009)
Martinet B.: Régularisation d’ inéquations variationelles par approximations succesives. Rev. Fr. Inf. Rech. Opér. 2, 154–159 (1970)
Németh S.Z.: Five kinds of monotone vector fields. Pure Math. Appl. 9, 417–428 (1998)
Németh S.Z.: Geodesic monotone vector fields. Lobachevskii J. Math. 5, 13–28 (1999)
Németh S.Z.: Monotonicity of the complementary vector field of a nonexpansive map. Acta Math. Hung. 84, 189–197 (1999)
Németh S.Z.: Monotone vector fields. Publ. Math. Debrecen 54, 437–449 (1999)
Németh S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52, 1491–1498 (2003)
Quiroz E.A.P., Quispe E.M., Oliveira P.R.: Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds. J. Math. Anal. Appl. 341, 467–477 (2008)
Quiroz E.A.P., Quispe E.M., Oliveira P.R.: Proximal point methods for quasiconvex and convex functions with Bregman distances on manifolds. J. Convex Anal. 16, 49–69 (2009)
Rockafellar R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Sakai, T.: Riemannian geometry. In: Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence (1996)
Schaible S., Yao J.C., Zeng L.C.: A proximal method for pseudomonotone type variational-like inequalities. Taiwan. J. Math. 10, 497–513 (2006)
Solodov M.V., Svaiter B.F.: Error bounds for proximal point subproblems and associated inexact proximal point algorithms. Math. Program. 88, 371–389 (2000)
Wang J.H., López G., Martín-Márquez V., Li C.: Monotone and accretive vector fields on Riemannian manifolds. J. Optim Theory Appl. 146, 691–708 (2010)
Wu K.Q., Huang N.J.: The generalized f-projection operator and set-valued variational inequalities in Banach spaces. Nonlinear Anal. TMA 71, 2481–2490 (2009)
Yamashita N., Fukushima M.: Equivalent uncontrained minimization and global error bounds for variational inequality problems. SIAM J. Control Optim. 35, 273–284 (1997)
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Tang, Gj., Zhou, Lw. & Huang, Nj. The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds. Optim Lett 7, 779–790 (2013). https://doi.org/10.1007/s11590-012-0459-7
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DOI: https://doi.org/10.1007/s11590-012-0459-7