TY - JOUR AB - Weak convergence of inertial iterative method for solving variational inequalities is the focus of this paper. The cost function is assumed to be non-Lipschitz and monotone. We propose a projection-type method with inertial terms and give weak convergence analysis under appropriate conditions. Some test results are performed and compared with relevant methods in the literature to show the efficiency and advantages given by our proposed methods. AU - Shehu, Yekini AU - Iyiola, Olaniyi S. ID - 7577 IS - 1 JF - Applicable Analysis SN - 0003-6811 TI - Weak convergence for variational inequalities with inertial-type method VL - 101 ER - TY - JOUR AB - In this paper, we consider reflected three-operator splitting methods for monotone inclusion problems in real Hilbert spaces. To do this, we first obtain weak convergence analysis and nonasymptotic O(1/n) convergence rate of the reflected Krasnosel'skiĭ-Mann iteration for finding a fixed point of nonexpansive mapping in real Hilbert spaces under some seemingly easy to implement conditions on the iterative parameters. We then apply our results to three-operator splitting for the monotone inclusion problem and consequently obtain the corresponding convergence analysis. Furthermore, we derive reflected primal-dual algorithms for highly structured monotone inclusion problems. Some numerical implementations are drawn from splitting methods to support the theoretical analysis. AU - Iyiola, Olaniyi S. AU - Enyi, Cyril D. AU - Shehu, Yekini ID - 9469 JF - Optimization Methods and Software SN - 1055-6788 TI - Reflected three-operator splitting method for monotone inclusion problem ER - TY - JOUR AB - In this paper, we present two new inertial projection-type methods for solving multivalued variational inequality problems in finite-dimensional spaces. We establish the convergence of the sequence generated by these methods when the multivalued mapping associated with the problem is only required to be locally bounded without any monotonicity assumption. Furthermore, the inertial techniques that we employ in this paper are quite different from the ones used in most papers. Moreover, based on the weaker assumptions on the inertial factor in our methods, we derive several special cases of our methods. Finally, we present some experimental results to illustrate the profits that we gain by introducing the inertial extrapolation steps. AU - Izuchukwu, Chinedu AU - Shehu, Yekini ID - 9234 IS - 2 JF - Networks and Spatial Economics KW - Computer Networks and Communications KW - Software KW - Artificial Intelligence SN - 1566-113X TI - New inertial projection methods for solving multivalued variational inequality problems beyond monotonicity VL - 21 ER - TY - JOUR AB - The paper introduces an inertial extragradient subgradient method with self-adaptive step sizes for solving equilibrium problems in real Hilbert spaces. Weak convergence of the proposed method is obtained under the condition that the bifunction is pseudomonotone and Lipchitz continuous. Linear convergence is also given when the bifunction is strongly pseudomonotone and Lipchitz continuous. Numerical implementations and comparisons with other related inertial methods are given using test problems including a real-world application to Nash–Cournot oligopolistic electricity market equilibrium model. AU - Shehu, Yekini AU - Iyiola, Olaniyi S. AU - Thong, Duong Viet AU - Van, Nguyen Thi Cam ID - 8817 IS - 2 JF - Mathematical Methods of Operations Research SN - 1432-2994 TI - An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems VL - 93 ER - TY - JOUR AB - We consider inertial iteration methods for Fermat–Weber location problem and primal–dual three-operator splitting in real Hilbert spaces. To do these, we first obtain weak convergence analysis and nonasymptotic O(1/n) convergence rate of the inertial Krasnoselskii–Mann iteration for fixed point of nonexpansive operators in infinite dimensional real Hilbert spaces under some seemingly easy to implement conditions on the iterative parameters. One of our contributions is that the convergence analysis and rate of convergence results are obtained using conditions which appear not complicated and restrictive as assumed in other previous related results in the literature. We then show that Fermat–Weber location problem and primal–dual three-operator splitting are special cases of fixed point problem of nonexpansive mapping and consequently obtain the convergence analysis of inertial iteration methods for Fermat–Weber location problem and primal–dual three-operator splitting in real Hilbert spaces. Some numerical implementations are drawn from primal–dual three-operator splitting to support the theoretical analysis. AU - Iyiola, Olaniyi S. AU - Shehu, Yekini ID - 9315 IS - 2 JF - Results in Mathematics SN - 1422-6383 TI - New convergence results for inertial Krasnoselskii–Mann iterations in Hilbert spaces with applications VL - 76 ER - TY - JOUR AB - In this paper, we propose a new iterative method with alternated inertial step for solving split common null point problem in real Hilbert spaces. We obtain weak convergence of the proposed iterative algorithm. Furthermore, we introduce the notion of bounded linear regularity property for the split common null point problem and obtain the linear convergence property for the new algorithm under some mild assumptions. Finally, we provide some numerical examples to demonstrate the performance and efficiency of the proposed method. AU - Ogbuisi, Ferdinard U. AU - Shehu, Yekini AU - Yao, Jen Chih ID - 9365 JF - Optimization SN - 0233-1934 TI - Convergence analysis of new inertial method for the split common null point problem ER - TY - JOUR AB - This paper aims to obtain a strong convergence result for a Douglas–Rachford splitting method with inertial extrapolation step for finding a zero of the sum of two set-valued maximal monotone operators without any further assumption of uniform monotonicity on any of the involved maximal monotone operators. Furthermore, our proposed method is easy to implement and the inertial factor in our proposed method is a natural choice. Our method of proof is of independent interest. Finally, some numerical implementations are given to confirm the theoretical analysis. AU - Shehu, Yekini AU - Dong, Qiao-Li AU - Liu, Lu-Lu AU - Yao, Jen-Chih ID - 8196 JF - Optimization and Engineering SN - 1389-4420 TI - New strong convergence method for the sum of two maximal monotone operators VL - 22 ER - TY - JOUR AB - In this paper, we introduce a relaxed CQ method with alternated inertial step for solving split feasibility problems. We give convergence of the sequence generated by our method under some suitable assumptions. Some numerical implementations from sparse signal and image deblurring are reported to show the efficiency of our method. AU - Shehu, Yekini AU - Gibali, Aviv ID - 7925 JF - Optimization Letters SN - 1862-4472 TI - New inertial relaxed method for solving split feasibilities VL - 15 ER - TY - JOUR AB - We consider the monotone variational inequality problem in a Hilbert space and describe a projection-type method with inertial terms under the following properties: (a) The method generates a strongly convergent iteration sequence; (b) The method requires, at each iteration, only one projection onto the feasible set and two evaluations of the operator; (c) The method is designed for variational inequality for which the underline operator is monotone and uniformly continuous; (d) The method includes an inertial term. The latter is also shown to speed up the convergence in our numerical results. A comparison with some related methods is given and indicates that the new method is promising. AU - Shehu, Yekini AU - Li, Xiao-Huan AU - Dong, Qiao-Li ID - 6593 JF - Numerical Algorithms SN - 1017-1398 TI - An efficient projection-type method for monotone variational inequalities in Hilbert spaces VL - 84 ER - TY - JOUR AB - The projection methods with vanilla inertial extrapolation step for variational inequalities have been of interest to many authors recently due to the improved convergence speed contributed by the presence of inertial extrapolation step. However, it is discovered that these projection methods with inertial steps lose the Fejér monotonicity of the iterates with respect to the solution, which is being enjoyed by their corresponding non-inertial projection methods for variational inequalities. This lack of Fejér monotonicity makes projection methods with vanilla inertial extrapolation step for variational inequalities not to converge faster than their corresponding non-inertial projection methods at times. Also, it has recently been proved that the projection methods with vanilla inertial extrapolation step may provide convergence rates that are worse than the classical projected gradient methods for strongly convex functions. In this paper, we introduce projection methods with alternated inertial extrapolation step for solving variational inequalities. We show that the sequence of iterates generated by our methods converges weakly to a solution of the variational inequality under some appropriate conditions. The Fejér monotonicity of even subsequence is recovered in these methods and linear rate of convergence is obtained. The numerical implementations of our methods compared with some other inertial projection methods show that our method is more efficient and outperforms some of these inertial projection methods. AU - Shehu, Yekini AU - Iyiola, Olaniyi S. ID - 8077 JF - Applied Numerical Mathematics SN - 0168-9274 TI - Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence VL - 157 ER - TY - JOUR AB - 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. AU - Shehu, Yekini AU - Gibali, Aviv AU - Sagratella, Simone ID - 7161 JF - Journal of Optimization Theory and Applications SN - 0022-3239 TI - Inertial projection-type methods for solving quasi-variational inequalities in real Hilbert spaces VL - 184 ER - TY - JOUR AB - It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators in Banach spaces. We obtain weak and strong convergence results for the zeros of the sum of maximal monotone and Lipschitz continuous monotone operators in Banach spaces. Many already studied problems in the literature can be considered as special cases of this paper. AU - Shehu, Yekini ID - 6596 IS - 4 JF - Results in Mathematics SN - 1422-6383 TI - Convergence results of forward-backward algorithms for sum of monotone operators in Banach spaces VL - 74 ER - TY - JOUR AB - The main contributions of this paper are the proposition and the convergence analysis of a class of inertial projection-type algorithm for solving variational inequality problems in real Hilbert spaces where the underline operator is monotone and uniformly continuous. We carry out a unified analysis of the proposed method under very mild assumptions. In particular, weak convergence of the generated sequence is established and nonasymptotic O(1 / n) rate of convergence is established, where n denotes the iteration counter. We also present some experimental results to illustrate the profits gained by introducing the inertial extrapolation steps. AU - Shehu, Yekini AU - Iyiola, Olaniyi S. AU - Li, Xiao-Huan AU - Dong, Qiao-Li ID - 7000 IS - 4 JF - Computational and Applied Mathematics SN - 2238-3603 TI - Convergence analysis of projection method for variational inequalities VL - 38 ER -