Iterative method with inertial for variational inequalities in Hilbert spaces
Tóm tắt
Strong convergence property for Halpern-type iterative method with inertial terms for solving variational inequalities in real Hilbert spaces is investigated under mild assumptions in this paper. Our proposed method requires only one projection onto the feasible set per iteration, the underline operator is monotone and uniformly continuous which is more applicable than most existing methods for which strong convergence is achieved and our method includes the inertial extrapolation step which is believed to increase the rate of convergence. Numerical comparisons of our proposed method with some other related methods in the literature are given.
Tài liệu tham khảo
Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14, 773–782 (2004)
Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)
Attouch, H., Goudon, X., Redont, P.: The heavy ball with friction. I. The continuous dynamical system. Commun. Contemp. Math. 2(1), 1–34 (2000)
Attouch, H., Czarnecki, M.O.: Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Differ. Equ. 179(1), 278–310 (2002)
Attouch, H., Peypouquet, J., Redont, P.: A dynamical approach to an inertial forward–backward algorithm for convex minimization. SIAM J. Optim. 24, 232–256 (2014)
Apostol, R.Y., Grynenko, A.A., Semenov, V.V.: Iterative algorithms for monotone bilevel variational inequalities. J. Comp. Appl. Math. 107, 3–14 (2012)
Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)
Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York (2011)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)
Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas–Rachford splitting for monotone inclusion. Appl. Math. Comput. 256, 472–487 (2015)
Bot, R.I., Csetnek, E.R.: An inertial alternating direction method of multipliers. Minimax Theory Appl. 1, 29–49 (2016)
Bot, R.I., Csetnek, E.R.: An inertial forward–backward–forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algoritm. 71, 519–540 (2016)
Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics, vol. 2057. Springer, Berlin (2012)
Ceng, L.C., Hadjisavvas, N., Wong, N.-C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Global Optim. 46, 635–646 (2010)
Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Methods Softw. 26, 827–845 (2011)
Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)
Chen, C., Chan, R.H., Ma, S., Yang, J.: Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM J. Imaging Sci. 8, 2239–2267 (2015)
Denisov, S., Semenov, V., Chabak, L.: Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern. Syst. Anal. 51, 757–765 (2015)
Glowinski, R., Lions, J.-L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)
Halpern, B.: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 73, 957–961 (1967)
Harker, P.T., Pang, J.-S.: A damped-Newton method for the linear complementarity problem. In: Allgower, G., Georg, K. (eds.) Computational Solution of Nonlinear Systems of Equations, Lectures in Applied Mathematics, vol. 26, pp. 265–284. AMS, Providence (1990)
He, Y.R.: A new double projection algorithm for variational inequalities. J. Comput. Appl. Math. 185, 166–173 (2006)
He, B.-S., Yang, Z.-H., Yuan, X.-M.: An approximate proximal-extragradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300, 362–374 (2004)
Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66, 75–96 (2017)
Iusem, A.N., Gárciga Otero, R.: Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces. Numer. Funct. Anal. Optim. 22, 609–640 (2001)
Iusem, A.N., Nasri, M.: Korpelevich’s method for variational inequality problems in Banach spaces. J. Global Optim. 50, 59–76 (2011)
Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997)
Kanzow, C., Shehu, Y.: Strong convergence of a double projection-type method for monotone variational inequalities in Hilbert Spaces. J. Fixed Point Theory Appl. 20, 51 (2018)
Khobotov, E.N.: Modification of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Math. Phys. 27, 120–127 (1989)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ékon. Mat. Metody 12, 747–756 (1976)
Kraikaew, R., Saejung, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 163, 399–412 (2014)
Lorenz, D.A., Pock, T.: An inertial forward–backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)
Maingé, P.E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007)
Maingé, P.E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 344, 876–887 (2008)
Maingé, P.-E., Gobinddass, M.L.: Convergence of one-step projected gradient methods for variational inequalities. J. Optim. Theory Appl. 171, 146–168 (2016)
Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)
Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Global Optim. 61, 193–202 (2015)
Marcotte, P.: Applications of Khobotov’s algorithm to variational and network equlibrium problems. Inf. Syst. Oper. Res. 29, 258–270 (1991)
Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006)
Ochs, P., Brox, T., Pock, T.: iPiasco: inertial proximal algorithm for strongly convex optimization. J. Math. Imaging Vis. 53, 171–181 (2015)
Polyak, B.T.: Some methods of speeding up the convergence of iterarive methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 1–17 (1964)
Shehu, Y., Iyiola, O.S.: Strong convergence result for monotone variational inequalities. Numer. Algoritm. 76, 259–282 (2017)
Shehu, Y., Iyiola, O.S.: Iterative algorithms for solving fixed point problems and variational inequalities with uniformly continuous monotone operators. Numer. Algoritm. 79, 529–553 (2018)
Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)
Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)
Xu, H.K.: Iterative algorithm for nonlinear operators. J. Lond. Math. Soc. 66(2), 1–17 (2002)