Modified Tseng's extragradient methods for variational inequality on Hadamard manifolds

References

• Facchinei F, Pang JS. Finite-dimensional variational inequalities and complementarity problems. Vol. II, Springer series in operations research. New York: Springer; 2003.
   Google Scholar
• Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their applications. Vol. 31, Classics in applied mathematics. Philadelphia (PA): Society for Industrial and Applied Mathematics (SIAM); 2000.
   Google Scholar
• Daniele P, Giannessi F, Maugeri A, editors. Equilibrium problems and variational models. Vol. 68, Nonconvex optimization and its applications. Norwell (MA): Kluwer Academic; 2003.
   Google Scholar
• Iusem AN, Pérez LRL. An extragradient-type algorithm for non-smooth variational inequalities. Optimization. 2000;48:309–332. doi: 10.1080/02331930008844508
   Google Scholar
• Iusem AN, Nasri M. Korpelevich's method for variational inequality problems in Banach spaces. J Global Optim. 2011;50:59–76. doi: 10.1007/s10898-010-9613-x
   Web of Science ®Google Scholar
• Iusem AN, Svaiter BF. A variant of Korpelevich's method for variational inequalities with a new search strategy. Optimization. 1997;42:309–321. doi: 10.1080/02331939708844365
   Google Scholar
• Chang XK, Liu SY, Zhao PJ, et al. A generalization of linearized alternating direction method of multipliers for solving two-block separable convex programming. J Comput Appl Math. 2019;357:251–272. doi: 10.1016/j.cam.2019.02.028
   Web of Science ®Google Scholar
• Chang XK, Liu SY, Zhao PJ, et al. Convergent prediction-correction-based ADMM for multi-block separable convex programming. J Comput Appl Math. 2018;335:270–288. doi: 10.1016/j.cam.2017.11.033
   Web of Science ®Google Scholar
• Tang GJ, Huang NJ. Strong convergence of an inexact projected subgradient method for mixed variational inequalities. Optimization. 2014;63:601–615. doi: 10.1080/02331934.2012.672420
   Web of Science ®Google Scholar
• Ceng LC, Ansari QH, Schaible S. Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems. J Global Optim. 2012;53:69–96. doi: 10.1007/s10898-011-9703-4
   Web of Science ®Google Scholar
• Ceng LC, Hadjisavvas N, Wong NC. Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J Global Optim. 2010;46:635–646. doi: 10.1007/s10898-009-9454-7
   Web of Science ®Google Scholar
• Ceng LC, Ansari QH, Wong MM, et al. Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems. Fixed Point Theory. 2012;13:403–422.
   Web of Science ®Google Scholar
• Ceng LC, Ansari QH, Yao JC. Relaxed extragradient iterative methods for variational inequalities. Appl Math Comput. 2011;218:1112–1123.
   Web of Science ®Google Scholar
• Ceng LC, Petruşel A, Yao JC, et al. Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces. Fixed Point Theory. 2018;19:487–501. doi: 10.24193/fpt-ro.2018.2.39
   Web of Science ®Google Scholar
• Ceng LC, Shang M. Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings. Optimization. 2019. DOI:10.1080/02331934.2019.1647203
   Web of Science ®Google Scholar
• Cruz Neto JXD, Ferreira OP, Pérez LRL, et al. Convex- and monotone-transformable mathematical programming problems and a proximal-like point method. J Global Optim. 2006;35:53–69. doi: 10.1007/s10898-005-6741-9
   Web of Science ®Google Scholar
• Quiroz EAP, Quispe EM, Oliveira PR. Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds. J Math Anal Appl. 2008;341:467–477. doi: 10.1016/j.jmaa.2007.10.010
   Web of Science ®Google Scholar
• Barani A, Pouryayevali MR. Invex sets and preinvex functions on Riemannian manifolds. J Math Anal Appl. 2007;328:767–779. doi: 10.1016/j.jmaa.2006.05.081
   Web of Science ®Google Scholar
• Barani A, Pouryayevali MR. Invariant monotone vector fields on Riemannian manifolds. Nonlinear Anal. 2009;70:1850–1861. doi: 10.1016/j.na.2008.02.085
   Web of Science ®Google Scholar
• Bento GC, Ferreira OP, Oliveira PR. Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds. Nonlinear Anal. 2010;73:564–572. doi: 10.1016/j.na.2010.03.057
   Web of Science ®Google Scholar
• Cruz Neto JXD, Ferreira OP, Lucambio Pérez LR. Contributions to the study of monotone vector fields. Acta Math Hungar. 2002;94:307–320. doi: 10.1023/A:1015643612729
   Web of Science ®Google Scholar
• Ferreira OP, Oliveira PR. Proximal point algorithm on Riemannian manifolds. Optimization. 2002;51:257–270. doi: 10.1080/02331930290019413
   Web of Science ®Google Scholar
• Ferreira OP, Pérez LRL, Németh SZ. Singularities of monotone vector fields and an extragradient-type algorithm. J Global Optim. 2005;31:133–151. doi: 10.1007/s10898-003-3780-y
   Web of Science ®Google Scholar
• Li C, López G, Martín-Márquez V. Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J Lond Math Soc. 2009;79(3):663–683. doi: 10.1112/jlms/jdn087
   Web of Science ®Google Scholar
• Li C, López G, Martín-Márquez V. Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J Math. 2010;14:541–559. doi: 10.11650/twjm/1500405806
   Web of Science ®Google Scholar
• Quiroz EAP, Oliveira PR. Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J Convex Anal. 2009;16:49–69.
   Web of Science ®Google Scholar
• Wang JH, López G, Martín-Márquez V, et al. Monotone and accretive vector fields on Riemannian manifolds. J Optim Theory Appl. 2010;146:691–708. doi: 10.1007/s10957-010-9688-z
   Web of Science ®Google Scholar
• Adler RL, Dedieu JP, Margulies JY, et al. Newton's method on Riemannian manifolds and a geometric model for the human spine. IMA J Numer Anal. 2002;22:359–390. doi: 10.1093/imanum/22.3.359
   Web of Science ®Google Scholar
• Dedieu JP, Priouret P, Malajovich G. Newton's method on Riemannian manifolds: convariant alpha theory. IMA J Numer Anal. 2003;23:395–419. doi: 10.1093/imanum/23.3.395
   Web of Science ®Google Scholar
• Edelman A, Arias TA, Smith ST. The geometry of algorithms with orthogonality constraints. SIAM J Matrix Anal Appl. 1999;20:303–353. doi: 10.1137/S0895479895290954
   Web of Science ®Google Scholar
• Ferreira OP, Svaiter BF. Kantorovich's theorem on Newton's method in Riemannian manifolds. J Complexity. 2002;18:304–329. doi: 10.1006/jcom.2001.0582
   Web of Science ®Google Scholar
• Li C, Wang J. Newton's method on Riemannian manifolds: smale's point estimate theory under the γ-condition. IMA J Numer Anal. 2006;26:228–251. doi: 10.1093/imanum/dri039
   Web of Science ®Google Scholar
• Azagra D, Ferrera J, López-Mesas F. Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds. J Funct Anal. 2005;220:304–361. doi: 10.1016/j.jfa.2004.10.008
   Web of Science ®Google Scholar
• Ledyaev YS, Zhu QJ. Nonsmooth analysis on smooth manifolds. Trans Amer Math Soc. 2007;359:3687–3732. doi: 10.1090/S0002-9947-07-04075-5
   Web of Science ®Google Scholar
• Hosseini S, Pouryayevali MR. Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds. Nonlinear Anal. 2011;74:3884–3895. doi: 10.1016/j.na.2011.02.023
   Web of Science ®Google Scholar
• Németh SZ. Variational inequalities on Hadamard manifolds. Nonlinear Anal. 2003;52:1491–1498. doi: 10.1016/S0362-546X(02)00266-3
   Web of Science ®Google Scholar
• Tang GJ, Zhou LW, Huang NJ. The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds. Optim Lett. 2013;7:779–790. doi: 10.1007/s11590-012-0459-7
   Web of Science ®Google Scholar
• Tang GJ, Huang NJ. Korpelevich's method for variational inequality problems on Hadamard manifolds. J Global Optim. 2012;54:493–509. doi: 10.1007/s10898-011-9773-3
   Web of Science ®Google Scholar
• Sakai T. Riemannian geometry. Vol. 149, Translations of mathematical monographs. Providence (RI): American Mathematical Society; 1996. Translated from the 1992 Japanese original by the author.
   Google Scholar
• Ansari QH, Babu F. Proximal point algorithm for inclusion problems in hadamard manifolds with applications. Optim Lett. 2019. DOI:10.1007/s11590-019-01483-0
   Web of Science ®Google Scholar
• Reich S. Strong convergence theorems for resolvents of accretive operators in Banach spaces. J Math Anal Appl. 1980;75:287–292. doi: 10.1016/0022-247X(80)90323-6
   Web of Science ®Google Scholar
• Reich S, Shafrir I. Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 1990;15:537–558. doi: 10.1016/0362-546X(90)90058-O
   Web of Science ®Google Scholar
• Thong DV, Hieu DV. Weak and strong convergence theorems for variational inequality problems. Numer Algorithms. 2018;78:1045–1060. doi: 10.1007/s11075-017-0412-z
   Web of Science ®Google Scholar
• Bridson MR, Haefliger A. Metric spaces of non-positive curvature. Vol. 319, Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences]. Berlin: Springer; 1999.
   Google Scholar