NEW INERTIAL IMPLICIT PROJECTION METHOD FOR SOLVING QUASI-VARIATIONAL INEQUALITIES IN REAL HILBERT SPACES
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https://doi.org/10.59287/ijanser.351Keywords:
Quasi-Variational İnequality, Inertial Extrapolation Step, Strong Monotonicity, Inertial Methods, ConvergenceAbstract
This paper introduces a new inertial implicit projection method to solve quasivariational inequalities with strongly monotone and Lipschitz continuous operators in real Hilbert spaces. We analysed the convergence of a method with varying stepsizes under suitable conditions and also discussed the complexity bound of the proposed algorithm.
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Antipin, A.S., Jacimovic, M., Mijajlovi, N.: Extragradient method for solving quasivariational inequalities. Optimization 67, 103112 (2018). [2] Aussel, D., Sagratella, S.: Sufficient conditions to compute any solution of a quasivariational inequality via a variational inequality. Math. Methods Oper. Res. 85, 318 (2017).
Attouch, H., Goudon, X., Redont, P.: The heavy ball with friction. I. The continuous dynamical system. Commun. Contemp. Math. 2, 134 (2000)
Attouch, H., Czarnecki, M.O.: Asymptotic control and stabilization of nonlinear oscillators with nonisolated equilibria. J. Diff. Equ. 179, 278310 (2002).
Attouch,H., Chbani,Z., Peypouquet,J., and Redont,P., Fast convergence of inertial dynamics and algorithms with asymptotic vanishing damping, Math. Program., to appear, DOI 10.1007/s10107-016-0992-8.
Alvarez, F., Attouch, H., An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Var. Anal., 9, 311(2001).
Attouch, H., Peypouquet, J., Redont, P.: A dynamical approach to an inertial forward-backward algorithm for convex minimization. SIAM J. Optim. 24, 232256(2014)
Attouch, H., Peypouquet, J.: The rate of convergence of Nesterovs accelerated forward-backward method is actually faster than 1/k2. SIAM J. Optim. 26, 18241834(2016)
Bensoussan, A., Lions,J. L.: Application des inequalities variationnelles en control eten stochastique, Paris: Dunod, (1978).
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183202 (2009).
Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial DouglasRachford splitting for monotone inclusion. Appl. Math. Comput. 256, 472487 (2015).
Bot, R.I., Csetnek, E.R.: An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms 71,519540(2016)
Berinde, V., Iterative approximation of fixed points: Lecture Notes in Mathematics 1912, Springer Berlin, (2007).
Chan, D., Pang,J.,: The generalized quasi-variational inequality problem, Math. Oper. Res., 7,211222(1982). [15] Cristescu,G., Lupsa,L.: Non-connected convexities and applications, Dordrecht: Kluwer Academic Publisher, (2002).
Cristescu,G., Gaianu,M.,: Shape properties of Noors convex sets, In: Proceed. Twelfth Symposium of Mathematics and its Applications, Timisoara, 2009, 113
Fichera, G.: Sul problema elastostatico di Signorini con ambigue condizioni al contorno (English translation: On Signorinis elastostatic problem with ambiguous boundary conditions). Atti Accad. Naz. Lincei VIII. Ser. Rend. Cl. Sci. Fis. Mat. Nat. 34, 138142(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 Signorinis problem with ambiguous boundary conditions). Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat., Sez. I VIII. Ser. 7, 91140(1964).
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry and NonExpansive Mappings. Marcel Dekker Inc, New York (1984)
Jacimovic,M., Mijajlovic, N.: On a continuous gradient-type method for solving quasi variational inequalities, Proc. Mont. Acad. Sci Arts., 19 , 1627(2011).
Jabeen, S., Noor, M. A., Noor, K. I.: Some new inertial projection methods for quasi variational Inequalities, Appl. Math. E Notes., 21, In press.(2021).
Jabeen,S., Bin-Mohsin, B., Noor,M.A., Noor,K.I.,Inertial projection methods for solving general quasi-variational inequalities, 6(2), 1075-1086(2020).
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980).
Mosco, U.: Implicit variational problems and quasi variational inequalities. Lecture Notes in Mathathematics, vol. 543. Springer, Berlin (1976).
Mijajlovic, N., Milojica, J., Noor, M. A., Gradient-type projection methods for quasi variational inequalities, Optim. Lett., 13, 18851896(2019).
Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasivariational inequalities. Discrete Contin. Dyn. Syst. 31(4), 13831396 (2011).
Noor, M.A., Oettli, W.: On general nonlinear complementarity problems and quasi equilibria. Le Mathematiche 49, 313331 (1994) [28] Noor, M.A.: An iterative scheme for a class of quasi variational inequalities. J. Math. Anal. Appl. 110, 463468 (1985)
Noor, M.A.: Quasi variational inequalities. Appl. Math. Lett. 1, 367370 (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, 917925 (2013)
Noor, M. A.: On general Quasi variational inequalities, J. King Saud Univ. Sci., 24,8188(2012)
Noor, M. A., Noor, K. I., Bnouhachem, A., On unified implicit method for variational inequalities, J. Comput. Appl. Math., 249 , 6973(2013).
Noor,M. A., Some developments in general variational inequalities, Appl. Math. Comput., 152,199277(2004).
Polyak, B.T.: Some methods of speeding up the convergence of iterative methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 791803 (1964).
Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. Acad. Sci Paris 258, 44134416 (1964).
Shehu,Y., Gibali,A., Sagratella,S.: Inertial Projection-type Methods for solving Quasi-Variational inequalities in Real Hilbert Spaces, J Optim Theory Appl,(2019).
Shehu, Y.: Convergence rate analysis of inertial KrasnoselskiiMann-type iteration with applications. Numer. Funct. Anal. Optim. 39, 10771091 (2018).
Su, W., Boyd, S., Cands, E.J.: A differential equation for modeling Nesterovs accelerated gradient method: theory and insights. Neural Inf. Process. Syst. 27, 25102518(2014).
