A class of nonlinear proximal point algorithms for variational inequality problems

References

• A. Auslender, M. Teboulle, and S. Ben-Tiba, A logarithmic-quadratic proximal method for variational inequalities, Comput. Optim. Appl. 12 (1999), pp. 31–40. doi: 10.1023/A:1008607511915
   Web of Science ®Google Scholar
• D. Bertsekas, Nonlinear Programming, Athena Scientific, Belmont, 1995.
   Google Scholar
• D. Bertsekas and E. Gafni, Projection method for variational inequalities with applications to the traffic assignment problem, Math. Program. Study 17 (1987), pp. 139–159. doi: 10.1007/BFb0120965
   Google Scholar
• A. Bnouhachem and M.A. Noor, A new predicto-corrector method for pseudomonotone nonlinear complementarity problems, Int. J. Comput. Math. 85 (2008), pp. 1023–1038.
   Web of Science ®Google Scholar
• A. Bnouhachem and M. Xu, An inexact LQP alternating direction method for solving a class of structured variational inequalities, Comput. Math. Appl. 67 (2014), pp. 671–680. doi: 10.1016/j.camwa.2013.12.010
   Web of Science ®Google Scholar
• G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems, Math. Program. 64 (1994), pp. 81–101. doi: 10.1007/BF01582566
   Web of Science ®Google Scholar
• Y. Censor and S. Zenios, Proximal minimization algorithm with D-functions, J. Optim. Theory Appl. 73 (1992), pp. 451–464. doi: 10.1007/BF00940051
   Web of Science ®Google Scholar
• S. Dafermos, Traffic equilibrium and variational inequalities, Transport. Sci. 14 (1980), pp. 42–54. doi: 10.1287/trsc.14.1.42
   Web of Science ®Google Scholar
• J. Eckstein, Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming, Math. Oper. Res. 18 (1993), pp. 202–226. doi: 10.1287/moor.18.1.202
   Web of Science ®Google Scholar
• J. Eckstein, Approximate iterations in Bregman-function-based proximal algorithms, Math. Program. Ser. A 83 (1998), pp. 113–123.
   Web of Science ®Google Scholar
• J. Eckstein and D. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program. 55 (1992), pp. 293–318. doi: 10.1007/BF01581204
   Web of Science ®Google Scholar
• F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003.
   Google Scholar
• M. Ferris and J. Pang, Engineering and economic applications of complementarity problems, SIAM Rev. 39 (1997), pp. 669–713. doi: 10.1137/S0036144595285963
   Web of Science ®Google Scholar
• X. Fu and A. Bnouhachem, An self-adaptive LQP method for constrained variational inequalities, Appl. Math. Comput. 189 (2007), pp. 1586–1600. doi: 10.1016/j.amc.2006.12.032
   Web of Science ®Google Scholar
• D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Comput. Math. Appl. 2 (1976), pp. 16–40. doi: 10.1016/0898-1221(76)90003-1
   Google Scholar
• D.R. Han, A hybrid entropic proximal decomposition method with self-adaptive strategy for solving variational inequality problems, Comput. Math. Appl. 55 (2008), pp. 101–115. doi: 10.1016/j.camwa.2007.03.015
   Web of Science ®Google Scholar
• P. Harker and J. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Program. 48 (1990), pp. 161–220. doi: 10.1007/BF01582255
   Web of Science ®Google Scholar
• B.S. He, A class of projection and contraction methods for monotone variational inequalities, Appl. Math. Optim. 35 (1997), pp. 69–76. doi: 10.1007/BF02683320
   Web of Science ®Google Scholar
• B.S. He and X.M. Yuan, The uniform framework of some proximal-based decomposition methods for monotone variational inequalities with separable structure, Pac. J. Optim. 8 (2012), pp. 817–844.
   Web of Science ®Google Scholar
• B.S. He, L.Z. Liao, D.R. Han, and H. Yang, A new inexact alternating direction method for monotone variational inequalities, Math. Program. 92 (2002), pp. 103–118. doi: 10.1007/s101070100280
   Web of Science ®Google Scholar
• B.S. He, X.M. Yuan, and J.Z. Zhang, Comparison of two kinds of prediction-correction methods for monotone variational inequalities, Comput. Optim. Appl. 27 (2004), pp. 247–267. doi: 10.1023/B:COAP.0000013058.17185.90
   Web of Science ®Google Scholar
• B.S. He, Y. Xu, and X.M. Yuan, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities, Comput. Optim. Appl. 35 (2006), pp. 19–46. doi: 10.1007/s10589-006-6442-4
   Web of Science ®Google Scholar
• B.S. He, X.L. Fu, and Z.K. Jiang, Proximal-point algorithm using a linear proximal term, J. Optim. Theory Appl. 141 (2009), pp. 299–319. doi: 10.1007/s10957-008-9493-0
   Web of Science ®Google Scholar
• M. Hestenes, Multiplier and gradient methods, J. Optim. Theory Appl. 4 (1969), pp. 303–320. doi: 10.1007/BF00927673
   Google Scholar
• G. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon 12 (1976), pp. 747–756.
   Google Scholar
• M. Kyono and M. Fukushima, Nonlinear proximal decomposition method for convex programming, J. Optim. Theory Appl. 106 (2000), pp. 357–372. doi: 10.1023/A:1004655531273
   Web of Science ®Google Scholar
• M. Li, A hybrid LQP-based method for structured variational inequalities, Int. J. Comput. Math. 89 (2012), pp. 1412–1425. doi: 10.1080/00207160.2012.688822
   Web of Science ®Google Scholar
• B. Martinet, Régularization d’ inéquations variationelles par approximations sucessives, Rev. Fr. Inform. Rech. Opér. 4 (1970), pp. 154–159.
   Google Scholar
• A. Nagurney and P. Ramanujam, Network Economics: A Variational Inequality Approach, Kluwer Academic, Boston, MA, 1993.
   Google Scholar
• M.A. Noor, K.I. Noor, and A. Bnouhachem, On a unified implicit method for variational inequalities, J. Comput. Appl. Math. 249 (2013), pp. 69–73. doi: 10.1016/j.cam.2013.02.011
   Web of Science ®Google Scholar
• N. Parikh and S. Boyd, Proximal algorithms, Found. Trends Optim. 1 (2013), pp. 123–231.
   Google Scholar
• M. Powell, A method for nonlinear constraints in minimization problems, in Optimization, R. Fletcher, ed, Academic Press, London, 1969, pp. 283–298.
   Google Scholar
• R. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1 (1976), pp. 97–116. doi: 10.1287/moor.1.2.97
   Google Scholar
• R. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), pp. 877–898. doi: 10.1137/0314056
   Web of Science ®Google Scholar
• M. Solodov and B. Svaiter, A unified framework for some inexact proximal point algorithms, Numer. Funct. Anal. Optim. 22 (2001), pp. 1013–1035. doi: 10.1081/NFA-100108320
   Web of Science ®Google Scholar
• M. Teboulle, Entropic proximal mappings with applications to nonlinear programming, Math. Oper. Res. 17 (1992), pp. 670–690. doi: 10.1287/moor.17.3.670
   Web of Science ®Google Scholar
• M. Teboulle, Convergence of proximal-like algorithms, SIAM J. Optim. 7 (1997), pp. 1069–1083. doi: 10.1137/S1052623495292130
   Web of Science ®Google Scholar
• P. Tseng, Alternating projection-proximal methods for convex programming and variational inequalities, SIAM J. Optim. 7 (1997), pp. 951–965. doi: 10.1137/S1052623495279797
   Web of Science ®Google Scholar
• X. Yuan and M. Li, An LQP-based decomposition method for solving a class of variational inequalities, SIAM J. Optim. 21 (2011), pp. 1309–1318. doi: 10.1137/070703557
   Web of Science ®Google Scholar