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 - 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 - 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 - 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
JF - Mathematical Methods of Operations Research
SN - 14322994
TI - An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems
ER -
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
JF - Applicable Analysis
SN - 0003-6811
TI - Weak convergence for variational inequalities with inertial-type method
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
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
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 - CONF
AB - A Valued Constraint Satisfaction Problem (VCSP) provides a common framework that can express a wide range of discrete optimization problems. A VCSP instance is given by a finite set of variables, a finite domain of labels, and an objective function to be minimized. This function is represented as a sum of terms where each term depends on a subset of the variables. To obtain different classes of optimization problems, one can restrict all terms to come from a fixed set Γ of cost functions, called a language.
Recent breakthrough results have established a complete complexity classification of such classes with respect to language Γ: if all cost functions in Γ satisfy a certain algebraic condition then all Γ-instances can be solved in polynomial time, otherwise the problem is NP-hard. Unfortunately, testing this condition for a given language Γ is known to be NP-hard. We thus study exponential algorithms for this meta-problem. We show that the tractability condition of a finite-valued language Γ can be tested in O(3‾√3|D|⋅poly(size(Γ))) time, where D is the domain of Γ and poly(⋅) is some fixed polynomial. We also obtain a matching lower bound under the Strong Exponential Time Hypothesis (SETH). More precisely, we prove that for any constant δ<1 there is no O(3‾√3δ|D|) algorithm, assuming that SETH holds.
AU - Kolmogorov, Vladimir
ID - 6725
SN - 1868-8969
T2 - 46th International Colloquium on Automata, Languages and Programming
TI - Testing the complexity of a valued CSP language
VL - 132
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 -