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A projection algorithm for set-valued variational inequalities on Hadamard manifolds

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Abstract

A projection algorithm is suggested for the set-valued variational inequality on Hadamard manifold. Under a pseudomonotone assumption on the underlying vector field, our method is proved to be globally convergent to a solution of variational inequality. This algorithm is a natural extension to the Hadamard manifold of the one introduced by Fang and He (Appl Math Comput, 217:9543–9511, 2011) in \({\mathbb {R}}^n\).

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References

  1. Allevi, E., Gnudi, A., Konnov, I.V.: The proximal point method for nonmonotone variational inequalities. Math. Methods Oper. Res. 63(3), 553–565 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

  3. Auslender, A., Teboulle, M.: Lagrangian duality and related multiplier methods for variational inequality problems. SIAM J. Optim. 10(4), 1097–1115 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bao, T.Q., Khanh, P.Q.: A projection-type algorithm for pseudomonotone nonlipschitzian multivalued variational inequalities. In: Eberhard, A., Hadjisavvas, N., Lus, D.T. (eds.) Generalized Convexity, Generalized Monotonicity and Applications, Proceedings of the 7th International Symposium on Generalized Convexity and Generalized Monotonicity, pp. 113–129. Springer, New York (2005)

  5. Barani, A., Pouryayevali, M.R.: Invex sets and preinvex functions on Riemannian manifolds. J. Math. Anal. Appl. 328, 767–779 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barani, A., Pouryayevali, M.R.: Invariant monotone vector fields on Riemannian manifolds. Nonlinear Anal. 70, 1850–1861 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bento, G.C., Cruz, J.X.D.: Neto, A subgradient method for multiobjective optimization on Riemannian manifolds. J. Optim. Theory Appl. 159, 125–137 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for solving the variational inequality problem in Euclidean space. Optimization 61(9), 1119–1132 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cruz Neto, J.X.D., Ferreira, O.P., Lucambio Perez, L.R.: Monotone point-to-set vector fields. Dedicated to Professor Constantin Udriste. Balkan J. Geom. Appl. 5, 69–79 (2000)

    MATH  MathSciNet  Google Scholar 

  10. Cruz Neto, J.X.D., Ferreira, O.P., Lucambio Perez, L.R.: Contributions to the study of monotone vector fields. Acta. Math. Hung. 94, 307–320 (2002)

    Article  MATH  Google Scholar 

  11. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, New York (2003)

    Google Scholar 

  12. Fang, C.J., He, Y.R.: A double projection algorithm for multi-valued variational inequalities and a unified framework of the method. Appl. Math. Comput. 217, 9543–9511 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fang, C.J., He, Y.R.: An extragradient method for generalized variational inequality. Pacif. J. Optim. 9(1), 47–59 (2013)

    MATH  MathSciNet  Google Scholar 

  14. Fang, C.J., Chen, S.L., Yang, C.D.: An algorithm for solving multi-valued variational inequality. J. Inequal. Appl. 2013, 218 (2013)

    Article  MathSciNet  Google Scholar 

  15. Fang, S.C., Peterson, E.L.: Generalized variational inequalities. J. Optim. Theory Appl. 38(3), 363–383 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  16. 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)

    Article  MATH  Google Scholar 

  17. Fukushima, M.: The primal Douglas–Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem. Math. Program. 72(1, Ser. A), 1–15 (1996)

  18. He, Y.R.: A new double projection algorithm for variational inequalities. J. Comput. Appl. Math. 185, 166–173 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Iusem, A.N., Lucambio Pérez, L.R.: An extragradient-type algorithm for nonsmooth variational inequalities. Optimization 48, 309–332 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18(4), 445–454 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  21. Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  22. 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)

    Article  MATH  MathSciNet  Google Scholar 

  23. Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mapping in Hadamard manifolds. Taiwan. J. Math. 14, 541–559 (2010)

    MATH  Google Scholar 

  24. 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

  25. Li, C., Yao, J.C.: 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), 2486–2514 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Li, F.L., He, Y.R.: An algorithm for generalized variational inequality with pseudomonotone mapping. J. Comput. Appl. Math. 228, 212–218 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. 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)

    Article  MATH  MathSciNet  Google Scholar 

  28. Németh, S.Z.: Geodesic monotone vector fields. Lobachevskii J. Math. 5, 13–28 (1999)

    MATH  MathSciNet  Google Scholar 

  29. Németh, S.Z.: Monotonity of the complementary vector field of a nonexpansive map. Acta Math. Hung. 84, 189–197 (1999)

    Article  MATH  Google Scholar 

  30. Németh, S.Z.: Monotone vector fields. Publ. Math. Debr. 54, 437–449 (1999)

    MATH  Google Scholar 

  31. Németh, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52, 1491–1498 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Quiroz, E.A.P., Quispe, E.M., Oliveira, P.R.: Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian. J. Math. Anal. Appl. 341, 467–477 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  34. Sakai, T.: Riemannian Geometry, Translations of Mathematical Monographs 149. American Mathematical Society, Providence (1996)

    Google Scholar 

  35. Salmon, G., Strodiot, J.-J., Nguyen, V.H.: A bundle method for solving variational inequalities. SIAM J. Optim. 14(3), 869–893 (2003)

    Article  MathSciNet  Google Scholar 

  36. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37(3), 765–776 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  37. Tang, G.J., Huang, N.J.: Korpelevich’s method for variational inequality problems on Hadamard manifolds. J. Glob. optim. 54, 493–509 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  38. Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic Publishers, Dordrecht (1994)

    Book  MATH  Google Scholar 

  39. Walter, R.: On the metric projections onto convex sets in Riemannian spaces. Arch. Math. XXV, 91–98 (1974)

  40. 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)

    Article  MATH  MathSciNet  Google Scholar 

  41. Wang, J.H., Li, C.: Convergence of the family of Euler–Halley type methods on Riemannian manifolds under the \(\gamma \)-condition. Taiwan. J. Math. 13, 585–606 (2009)

    MATH  Google Scholar 

  42. Wang, J.H., López, G.: Modified proximal point algorithms on Hadamard manifolds. Optimization 60(6), 697–708 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  43. Wang, Y.J., Xiu, N.H., Zhang, J.Z.: Modified extragradient method for variational inequalities and verification of solution existence. J. Optim. Theory Appl. 119(1), 167–183 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This work is partially supported by Natural Science Foundation Project of CQ CSTC (No. 2010BB9401), Scientific and Technological Research Program of Chongqing Municipal Education Commission (No. KJ110509), Foundation of Chongqing University of Posts and Telecommunications for the Scholars with Doctorate (No. A2012-04) and Basic and Advanced Research Project of CQ CSTC(No.cstc2014jcyjA00044).

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Authors and Affiliations

  1. Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, People’s Republic of China

    Chang-jie Fang & Sheng-lan Chen

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  1. Chang-jie Fang
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  2. Sheng-lan Chen
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Correspondence to Chang-jie Fang.

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Fang, Cj., Chen, Sl. A projection algorithm for set-valued variational inequalities on Hadamard manifolds. Optim Lett 9, 779–794 (2015). https://doi.org/10.1007/s11590-014-0785-z

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