A new hybrid cutting-projection algorithm for equilibrium and fixed point problems
Nội dung chính của bài viết
Tóm tắt
The paper proposes a novel hybrid cutting-projection method for solving equilibrium problems and fixed point problems. By constructing
specially cutting-halfspaces, the method only requires to solve a strongly
convex optimization program at each iteration without the extra-steps as
extragradient methods. The strongly convergence theorem is established
and some numerical examples are presented to illustrate its convergence.
Từ khóa
Equilibrium Problem, Fixed Point Problem, Hybrid Method, Projection Method
Chi tiết bài viết
Tài liệu tham khảo
[1] Anh, P. K. and Hieu, D. V., Parallel hybrid iterative methods for variational inequalities, equilibrium problems, and common fixed point problems, Vietnam Journal of Mathematics, 44(2) (2016), 351-374.
[2] Anh, P. N., A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems, The Bulletin of the Malaysian Mathematical Society, 1(1) (2013), 107-116.
[3] Bigi, G., Castellani, M., Pappalardo, M. and Passacantando, M., Nonlinear programming techniques for equilibria, Springer, 2019.
[4] Blum, E. and Oettli, W., From optimization and variational inequalities to equilibrium problems, Mathematics Student, 63 (1994) 123-145.
[5] Censor, Y., Gibali, A. and Reich, S., Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optimization Methods and Software, 26(4-5)(2011), 827-845.
[6] Combettes, P. L., and Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 6(1) (2005), 117-136.
[7] Daniele, P., Giannessi, F. and Maugeri, A., Equilibrium problems and variational models, Kluwer Academic Publishers, 2003.
[8] Dinh, B.V., Hung, P.G. and Muu, L.D., Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numerical Functional Analysis and Optimization, 35(5) (2014), 539-563.
[9] Fan, K. and Shisha, O., A minimax inequality and applications, Inequalities III, Academic Press New York, (1972), (103-113).
[10] Goebel, K. and Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, (1984).
[11] Goebel, K. and Kirk, W.A., Topics in Metric Fixed Point Theory,Cambridge University Press, (1990). 18 P.K. Quy and D.V. Hieu
[12] Hieu, D. V., A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space, Journal of the Korean Mathematical Society, 52(2) (2015), 373-388.
[13] Hieu, D.V., A new shrinking gradient-like projection method for equilibrium problems, Optimization, 66(12) (2017), 2291-2307.
[14] Hieu, D.V., An extension of hybrid method without extrapolation step to equilibrium problems, Journal of Industrial and Management Optimization, 13(4) (2017), 1723-1741.
[15] Konnov, I.V., Combined relaxation methods for variational inequalities, Springer, (2001).
[16] Korpelevich, G. M., The extragradient method for finding saddle points and other problems, Ekonomikai Matematicheskie Metody, 12 (1976), 747-756.
[17] Malitsky, Y. V. and Semenov, V. V., A hybrid method without extrapolation step for solving variational inequality problems. Journal of Global Optimization, 61(1) (2015), 193-202.
[18] Martinet, B., R´egularisation d’in´equations variationnelles par approximations successives, Mathematical Modelling and Numerical Analysis, 3 (1970), 154-159.
[19] Mordukhovich, B., Panicucci, B., Pappalardo, M. and Passacantando, M., Hybrid proximal methods for equilibrium problems, Optimization Letters, 6 (2012), 1535-1550.
[20] Muu, L.D. and Oettli, W., Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Analysis, 18(12) (1992), 1159-1166.
[21] Muu, L. D., Thanh, L. X. and Strodiot, J. J., A survey on solution approaches for the equilibrium problem defined by the Nikaido-Isoda-Fan inequality, Mathematics and Mathematical Sciences, 2(8) (2023), 67-85.
[22] Nadezhkina, N. and Takahashi, W., Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM Journal on Optimization, 16(4) (2006), 1230-1241.
[23] Nguyen, T. T. V., Strodiot, J. J. and Nguyen, V. H., Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space, Journal of Optimization Theory and Applications, 160(3) (2014), 809–831.
[24] Quoc, T.D., Muu, L.D. and Hien, N.V., Extragradient algorithms extended to equilibrium problems, Optimization, 57(6) (2008), 749-776.
[25] Rockafellar, R.T., Convex Analysis, Princeton University Press, (1970).
[26] Rockafellar, R.T., Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14(5) (1976), 877–898.
[27] Solodov, M. V. and Svaiter, B. F., A new projection method for variational inequality problems, SIAM Journal on Control and Optimization, 37(3) (1999), 765-776.
[28] Solodov, M. V. and Svaiter, B. F., Forcing strong convergence ofproximal point iterations in Hilbert space, Mathematical Programming, 87(1) (2000), 189-202.
[29] Strodiot, J. J., Nguyen, V. H. and Vuong, P. T., Strong convergence of two hybrid extragradient methods for solving equilibrium and fixed point problems, Vietnam Journal of Mathematics, 40(2) (2012), 371-389.
[30] Strodiot, J. J., Nguyen, T. T. V and Nguyen, V. H., A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems, Journal of Global Optimization, 56(2) (2013), 373-397.
[31] Yamada, I., The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms for Feasibility and Optimization and Their Applications, Elsevier, Amsterdam, (2001), 473–504.
[2] Anh, P. N., A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems, The Bulletin of the Malaysian Mathematical Society, 1(1) (2013), 107-116.
[3] Bigi, G., Castellani, M., Pappalardo, M. and Passacantando, M., Nonlinear programming techniques for equilibria, Springer, 2019.
[4] Blum, E. and Oettli, W., From optimization and variational inequalities to equilibrium problems, Mathematics Student, 63 (1994) 123-145.
[5] Censor, Y., Gibali, A. and Reich, S., Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optimization Methods and Software, 26(4-5)(2011), 827-845.
[6] Combettes, P. L., and Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 6(1) (2005), 117-136.
[7] Daniele, P., Giannessi, F. and Maugeri, A., Equilibrium problems and variational models, Kluwer Academic Publishers, 2003.
[8] Dinh, B.V., Hung, P.G. and Muu, L.D., Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numerical Functional Analysis and Optimization, 35(5) (2014), 539-563.
[9] Fan, K. and Shisha, O., A minimax inequality and applications, Inequalities III, Academic Press New York, (1972), (103-113).
[10] Goebel, K. and Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, (1984).
[11] Goebel, K. and Kirk, W.A., Topics in Metric Fixed Point Theory,Cambridge University Press, (1990). 18 P.K. Quy and D.V. Hieu
[12] Hieu, D. V., A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space, Journal of the Korean Mathematical Society, 52(2) (2015), 373-388.
[13] Hieu, D.V., A new shrinking gradient-like projection method for equilibrium problems, Optimization, 66(12) (2017), 2291-2307.
[14] Hieu, D.V., An extension of hybrid method without extrapolation step to equilibrium problems, Journal of Industrial and Management Optimization, 13(4) (2017), 1723-1741.
[15] Konnov, I.V., Combined relaxation methods for variational inequalities, Springer, (2001).
[16] Korpelevich, G. M., The extragradient method for finding saddle points and other problems, Ekonomikai Matematicheskie Metody, 12 (1976), 747-756.
[17] Malitsky, Y. V. and Semenov, V. V., A hybrid method without extrapolation step for solving variational inequality problems. Journal of Global Optimization, 61(1) (2015), 193-202.
[18] Martinet, B., R´egularisation d’in´equations variationnelles par approximations successives, Mathematical Modelling and Numerical Analysis, 3 (1970), 154-159.
[19] Mordukhovich, B., Panicucci, B., Pappalardo, M. and Passacantando, M., Hybrid proximal methods for equilibrium problems, Optimization Letters, 6 (2012), 1535-1550.
[20] Muu, L.D. and Oettli, W., Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Analysis, 18(12) (1992), 1159-1166.
[21] Muu, L. D., Thanh, L. X. and Strodiot, J. J., A survey on solution approaches for the equilibrium problem defined by the Nikaido-Isoda-Fan inequality, Mathematics and Mathematical Sciences, 2(8) (2023), 67-85.
[22] Nadezhkina, N. and Takahashi, W., Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM Journal on Optimization, 16(4) (2006), 1230-1241.
[23] Nguyen, T. T. V., Strodiot, J. J. and Nguyen, V. H., Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space, Journal of Optimization Theory and Applications, 160(3) (2014), 809–831.
[24] Quoc, T.D., Muu, L.D. and Hien, N.V., Extragradient algorithms extended to equilibrium problems, Optimization, 57(6) (2008), 749-776.
[25] Rockafellar, R.T., Convex Analysis, Princeton University Press, (1970).
[26] Rockafellar, R.T., Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14(5) (1976), 877–898.
[27] Solodov, M. V. and Svaiter, B. F., A new projection method for variational inequality problems, SIAM Journal on Control and Optimization, 37(3) (1999), 765-776.
[28] Solodov, M. V. and Svaiter, B. F., Forcing strong convergence ofproximal point iterations in Hilbert space, Mathematical Programming, 87(1) (2000), 189-202.
[29] Strodiot, J. J., Nguyen, V. H. and Vuong, P. T., Strong convergence of two hybrid extragradient methods for solving equilibrium and fixed point problems, Vietnam Journal of Mathematics, 40(2) (2012), 371-389.
[30] Strodiot, J. J., Nguyen, T. T. V and Nguyen, V. H., A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems, Journal of Global Optimization, 56(2) (2013), 373-397.
[31] Yamada, I., The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms for Feasibility and Optimization and Their Applications, Elsevier, Amsterdam, (2001), 473–504.