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Research Article

Customized Douglas-Rachford splitting methods for structured inverse variational inequality problems

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Pages 257-281 | Received 04 Jan 2023, Accepted 19 Oct 2023, Published online: 24 Nov 2023

References

  • F. Anceschi, A. Barbagallo, and S. Guarino Lo Bianco, Inverse tensor variational inequalities and applications, J. Optim. Theory. Appl. 196 (2023), pp. 570–589.
  • A. Barbagallo and S. Guarino Lo Bianco, A random time-dependent noncooperative equilibrium problem, Comput. Optim. Appl. 84 (2023), pp. 27–52.
  • A. Barbagallo and P. Mauro, Inverse variational inequality approach and applications, Numer Funct Anal Optim. 35 (2014), pp. 851–867.
  • A. Barbagallo and P. Mauro, A general quasi-variational problem of Cournot-Nash type and its inverse formulation, J. Optim. Theory. Appl. 170 (2016), pp. 476–492.
  • H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics, 2nd edn, Springer, Cham, 2017.
  • S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn. 3(1) (2011), pp. 1–122.
  • X.J. Cai, K. Guo, F. Jiang, K. Wang, Z.M. Wu, and D.R. Han, The developments of proximal point algorithms, J. Oper. Res. Soc. China 10 (2022), pp. 197–239.
  • X.J. Cai, D.R. Han, and X.M. Yuan, On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function, Comput. Optim. Appl. 66 (2017), pp. 39–73.
  • J.W. Chen, X.X. Ju, E. Köbis, and Y.C. Liou, Tikhonov type regularization methods for inverse mixed variational inequalities, Optimization 69 (2020), pp. 401–413.
  • J. Douglas and H.H. Rachford, On the numerical solution of heat conduction problems in two and three space variables, Trans. Am. Math. Soc. 82 (1956), pp. 421–439.
  • R. Glowinski, On alternating direction methods of multipliers: A historical perspective, in Modeling, Simulation and Optimization for Science and Technology, Springer, 2014, pp. 59–82.
  • D.R. Han, A survey on some recent developments of alternating direction method of multipliers, J. Oper. Res. Soc. China 10(1) (2022), pp. 1–52.
  • D.R. Han, H.J. He, H. Yang, and X.M. Yuan, A customized Douglas-Rachford splitting algorithm for separable convex minimization with linear constraints, Numerische Mathematik 127 (2014), pp. 167–200.
  • D.R. Han and X.M. Yuan, Local linear convergence of the alternating direction method of multipliers for quadratic programs, SIAM. J. Numer. Anal. 51 (2013), pp. 3446–3457.
  • B.S. He, Inexact implicit methods for monotone general variational inequalities, Math. Program. 86 (1999), pp. 199–217.
  • H.J. He and D.R. Han, A distributed Douglas-Rachford splitting method for multi-block convex minimization problems, Adv. Comput. Math. 42 (2016), pp. 27–53.
  • B.S. He, X.Z. He, and H.X. Liu, Solving a class of constrained 'black-box' inverse variational inequalities, Eur. J. Oper. Res. 204 (2010), pp. 391–401.
  • B.S. He, L.Z. Liao, and S.L. Wang, Self-adaptive operator splitting methods for monotone variational inequalities, Numerische Mathematik 94 (2003), pp. 715–737.
  • X.Z. He and H.X. Liu, Inverse variational inequalities with projection-based solution methods, Eur. J. Oper. Res. 208 (2011), pp. 12–18.
  • R. Hu and Y.P. Fang, Levitin-Polyak well-posedness by perturbations for the split inverse variational inequality problem, J. Fixed Point Theory Appl. 18 (2016), pp. 785–800.
  • R. Hu and Y.P. Fang, Well-posedness of the split inverse variational inequality problem, Bull. Malays. Math. Sci. Soc. 40 (2017), pp. 1733–1744.
  • Y.N. Jiang, X.J. Cai, and D.R. Han, Solving policy design problems: Alternating direction method of multipliers-based methods for structured inverse variational inequalities, Eur. J. Oper. Res. 280 (2020), pp. 417–427.
  • X.X. Ju, C.D. Li, X. He, and G. Feng, An inertial projection neural network for solving inverse variational inequalities, Neurocomputing 406 (2020), pp. 99–105.
  • X.X. Ju, C.D. Li, X. He, and G. Feng, A proximal neurodynamic model for solving inverse mixed variational inequalities, Neural. Netw. 138 (2021), pp. 1–9.
  • D.S. Kim, P.T. Vuong, and P.D. Khanh, Qualitative properties of strongly pseudomonotone variational inequalities, Optim. Lett. 10 (2016), pp. 1669–1679.
  • X. Li, X.S. Li, and N.J. Huang, A generalized f-projection algorithm for inverse mixed variational inequalities, Optim. Lett. 8 (2014), pp. 1063–1076.
  • P.L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM. J. Numer. Anal. 16 (1979), pp. 964–979.
  • X.P. Luo, Tikhonov regularization methods for inverse variational inequalities, Optim. Lett. 8 (2014), pp. 877–887.
  • Z.Q. Luo and P. Tseng, Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem, SIAM. J. Optim. 2 (1992), pp. 43–54.
  • X.P. Luo and J. Yang, Regularization and iterative methods for monotone inverse variational inequalities, Optim. Lett. 8 (2014), pp. 1261–1272.
  • M.A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004), pp. 199–277.
  • J.S. Pang, Error bounds in mathematical programming, Math. Program. 79 (1997), pp. 299–332.
  • J.S. Pang and L.Q. Qi, Nonsmooth equations: Motivation and algorithms, SIAM. J. Optim. 3 (1993), pp. 443–465.
  • J.S. Pang and J.C. Yao, On a generalization of a normal map and equation, SIAM J Control Optim. 33 (1995), pp. 168–184.
  • L. Scrimali, An inverse variational inequality approach to the evolutionary spatial price equilibrium problem, Optim. Eng. 13 (2012), pp. 375–387.
  • P. Tseng, On linear convergence of iterative methods for the variational inequality problem, J. Comput. Appl. Math. 60 (1995), pp. 237–252.
  • P. Tseng, Approximation accuracy, gradient methods, and error bound for structured convex optimization, Math. Program. 125 (2010), pp. 263–295.
  • P. Tseng and S. Yun, A coordinate gradient descent method for linearly constrained smooth optimization and support vector machines training, Comput. Optim. Appl. 47 (2010), pp. 179–206.
  • D. Van Hieu, Y.J. Cho, Y.B. Xiao, and P. Kumam, Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional hilbert spaces, Vietnam J. Math. 49 (2021), pp. 1165–1183.
  • R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, 1966.
  • P.T. Vuong, X.Z. He, and D.V. Thong, Global exponential stability of a neural network for inverse variational inequalities, J. Optim. Theory. Appl. 190 (2021), pp. 915–930.
  • K. Wang and J. Desai, On the convergence rate of the augmented Lagrangian-based parallel splitting method, Optim. Methods Softw. 34 (2019), pp. 278–304.
  • H.K. Xu, S. Dey, and V. Vetrivel, Notes on a neural network approach to inverse variational inequalities, Optimization 70 (2021), pp. 901–910.
  • J.F. Yang, Dynamic power price problem: An inverse variational inequality approach, J. Ind. Manag. Optim. 4 (2008), pp. 673–684.
  • W.H. Yang and D.R. Han, Linear convergence of the alternating direction method of multipliers for a class of convex optimization problems, SIAM. J. Numer. Anal.54 (2016), pp. 625–640.
  • L.L. Yin, H.W. Liu, and J. Yang, Modified golden ratio algorithms for pseudomonotone equilibrium problems and variational inequalities, Appl. Math. 67 (2022), pp. 273–296.
  • X.J. Zou, D.W. Gong, L.P. Wang, and Z.Y. Chen, A novel method to solve inverse variational inequality problems based on neural networks, Neurocomputing 173 (2016), pp. 1163–1168.

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