Iterative method with inertial for variational inequalities in Hilbert spaces

A Parallel-Viscosity-Type Subgradient Extragradient-Line Method for Finding the Common Solution of Variational Inequality Problems Applied to Image Restoration Problems

“…The variational inequality theory is an important tool based on studying a wide class of problems-unilateral and equilibrium problems arising in structural analysis, economics, optimization, operations research, and engineering sciences (see [2][3][4][5][6][7] and the references therein). Several algorithms have been improved for solving variational inequality and related optimization problems (see [6,[8][9][10][11][12][13][14][15] and the references therein). It is well known that x is the solution of the VIP (5) if and only if x is the fixed point of the mapping P C (I − rA), r > 0 (see [4] for details)…”

Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

“…Motivated by the works of Yang et al [34] and Thong et al [35] and the current research interest in this direction, we propose two new inertial-type algorithms for solving the VIP (1) based on the TEGM and Moudafi's viscosity scheme which does not require a prior knowledge of the Lipschitz constant of the monotone operator. The inertial term ( − ) − α x x n n n 1 introduced can be regarded as a procedure for speeding up the convergence properties (see, for example, [22,23,33,[36][37][38][39]). The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, which is easy to compute.…”

“…Variational inequality theory is an important tool in economics, engineering, mathematical programming, transportation, and in other fields (see, for example, [1][2][3][4][5][6][7][8]). Many numerical methods have been constructed for solving variational inequalities and related optimization problems, see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] and references therein.…”

A modified contraction method for solving certain class of split monotone variational inclusion problems with application

“…Moreover, as we shall see in Sect. 4, the proof of the strong convergence of our method does not rely on the "Two Cases Approach" used by Ezeora and Izuchukwu (2019) and widely used in many papers to guarantee strong convergence (see, for example, Cholamjiak et al 2018;Dong et al 2017;Izuchukwu et al 2018;Shehu et al 2019a;Suantai et al 2019;Senakka and Cholamjiak 2016;Hieu 2017, 2018;Gibali et al 2019;Shehu et al 2019b;Shehu and Cholamjiak 2019;Thong and Cholamjiak 2019;Tuyen et al 2019 and the references therein). We also give some numerical illustrations of the proposed method in comparison with Algorithm (1.8) and Algorithm (1.14) to further show the applicability and efficiency of our method.…”

“…,Ezeora and Izuchukwu (2019),Khan et al (2018),Moudafi (2011),Tian and Jiang (2019), andTian and Jiang (2017) where {λ n } depends on the knowledge of L. This makes our algorithm generally more applicable than those inCholamjiak et al (2018),Dong et al (2017),Ezeora and Izuchukwu (2019),Khan et al (2018) andMoudafi (2011).• As we shall see in our convergence analysis, the proof of the strong convergence of Algorithm 3.2 (that is, the proof of Theorem 4.3) does not rely on the usual "Two Cases Approach" (Case 1 and Case 2) which usually used in numerous papers for solving optimization problems(Cholamjiak et al 2018;Dong et al 2017;Ezeora and Izuchukwu 2019;Izuchukwu et al 2018;Senakka and Cholamjiak 2016;Shehu et al 2019a;Suantai et al 2019; Hieu 2017, 2018;Gibali et al 2019;Shehu et al 2019a;Shehu and Cholamjiak 2019;Thong and Cholamjiak 2019;Tuyen et al 2019)…”