Abstract
In this paper, we introduce an inertial projection-type method with different updating strategies for solving quasi-variational inequalities with strongly monotone and Lipschitz continuous operators in real Hilbert spaces. Under standard assumptions, we establish different strong convergence results for the proposed algorithm. Primary numerical experiments demonstrate the potential applicability of our scheme compared with some related methods in the literature.
Similar content being viewed by others
References
Fichera, G.: Sul problema elastostatico di Signorini con ambigue condizioni al contorno (English translation: “On Signorini’s elastostatic problem with ambiguous boundary conditions”). Atti Accad. Naz. Lincei VIII. Ser. Rend. Cl. Sci. Fis. Mat. Nat. 34, 138–142 (1963)
Fichera, G.: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno (English translation: “Elastostatic problems with unilateral constraints: the Signorini’s problem with ambiguous boundary conditions”). Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat., Sez. I VIII. Ser. 7, 91–140 (1964)
Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. Acad. Sci Paris 258, 4413–4416 (1964)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
Antipin, A.S., Jaćimović, M., Mijajlovi, N.: Extragradient method for solving quasivariational inequalities. Optimization 67, 103–112 (2018)
Mosco, U.: Implicit variational problems and quasi variational inequalities. Lecture Notes in Mathathematics, vol. 543. Springer, Berlin (1976)
Noor, M.A.: An iterative scheme for a class of quasi variational inequalities. J. Math. Anal. Appl. 110, 463–468 (1985)
Noor, M.A.: Quasi variational inequalities. Appl. Math. Lett. 1, 367–370 (1988)
Noor, M.A., Noor, K.I., Khan, A.G.: Some iterative schemes for solving extended general quasi variational inequalities. Appl. Math. Inf. Sci. 7, 917–925 (2013)
Mijajlović, N., Jaćimović, M., Noor, M.A.: Gradient-type projection methods for quasi-variational inequalities. Optim. Lett. 13, 1885–1896 (2019)
Aussel, D., Sagratella, S.: Sufficient conditions to compute any solution of a quasivariational inequality via a variational inequality. Math. Methods Oper. Res. 85, 3–18 (2017)
Facchinei, F., Kanzow, C., Karl, S., Sagratella, S.: The semismooth Newton method for the solution of quasi-variational inequalities. Comput. Optim. Appl. 62, 85–109 (2015)
Facchinei, F., Kanzow, C., Sagratella, S.: Solving quasi-variational inequalities via their KKT conditions. Math. Program. 144, 369–412 (2014)
Latorre, V., Sagratella, S.: A canonical duality approach for the solution of affine quasi-variational inequalities. J. Global Optim. 64, 433–449 (2016)
Attouch, H., Goudon, X., Redont, P.: The heavy ball with friction. I. The continuous dynamical system. Commun. Contemp. Math. 2, 1–34 (2000)
Attouch, H., Czarnecki, M.O.: Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Diff. Equ. 179, 278–310 (2002)
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 (2003)
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)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)
Maingé, P.E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 344, 876–887 (2008)
Polyak, B.T.: Some methods of speeding up the convergence of iterative methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 791–803 (1964)
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)
Lorenz, D.A., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)
Ochs, P., Brox, T., Pock, T.: iPiasco: inertial proximal algorithm for strongly convex optimization. J. Math. Imaging Vis. 53, 171–181 (2015)
Boţ, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas–Rachford splitting for monotone inclusion. Appl. Math. Comput. 256, 472–487 (2015)
Shehu, Y.: Convergence rate analysis of inertial Krasnoselskii–Mann-type iteration with applications. Numer. Funct. Anal. Optim. 39, 1077–1091 (2018)
Boţ, R.I., Csetnek, E.R.: An inertial alternating direction method of multipliers. Minimax Theory Appl. 1, 29–49 (2016)
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)
Boţ, R.I., Csetnek, E.R.: An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms 71, 519–540 (2016)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry and Non-Expansive Mappings. Marcel Dekker Inc, New York (1984)
Noor, M.A., Oettli, W.: On general nonlinear complementarity problems and quasi equilibria. Matematiche (Catania) 49, 313–331 (1994)
Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)
Ryazantseva, I.P.: First-order methods for certain quasi-variational inequalities in a Hilbert space. Comput. Math. Math. Phys. 47, 183–190 (2007)
Antipin, A.S., Jaćimović, M., Mijajlovi, N.: A second-order iterative method for solving quasi-variational inequalities. Comput. Math. Math. Phys. 53, 258–264 (2013)
Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities. Discrete Contin. Dyn. Syst. 31, 1383–1396 (2011)
Attouch, H., Peypouquet, J.: The rate of convergence of Nesterov’s accelerated forward-backward method is actually faster than \(1/k^2\). SIAM J. Optim. 26, 1824–1834 (2016)
Facchinei, F., Kanzow, C., Sagratella, S.: QVILIB: a library of quasi-variational inequality test problems. Pac. J. Optim. 9, 225–250 (2013)
Acknowledgements
We are grateful to the anonymous referees and editor whose insightful comments helped to considerably improve an earlier version of this paper. The research of the first author is supported by an ERC Grant from the Institute of Science and Technology (IST).
Author information
Authors and Affiliations
Corresponding author
Additional information
Radu Ioan Boţ.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shehu, Y., Gibali, A. & Sagratella, S. Inertial Projection-Type Methods for Solving Quasi-Variational Inequalities in Real Hilbert Spaces. J Optim Theory Appl 184, 877–894 (2020). https://doi.org/10.1007/s10957-019-01616-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-019-01616-6