Inertial Step Research Articles

A Unified Inertial Iterative Approach for General Quasi Variational Inequality with Application

In this paper, we design two inertial iterative methods involving one and two inertial steps for investigating a general quasi-variational inequality in a real Hilbert space. We establish an existence result and a non-trivial example is furnished to substantiate our theoretical findings. We discuss the convergence of the inertial iterative algorithms to approximate the solution of a general quasi-variational inequality. Finally, we apply an inertial iterative scheme with two inertial steps to investigate a delay differential equation. The results presented herein can be seen as substantial generalizations of some known results.

New self-adaptive methods with double inertial steps for solving splitting monotone variational inclusion problems with applications

In this paper, without the co-coerciveness assumption of the associated mappings, we show the iterative sequences generated by our new algorithms converge strongly to a solution of the splitting monotone variational inclusion problem in infinite dimensional real Hilbert spaces. The step sizes of the new algorithms are updated per iteration by a simple calculation without knowing the prior information of the operator norm. Some parameters are relaxed to enlarge the value range of the corresponding step sizes. Double inertial extrapolation steps are incorporated in the new algorithms to accelerate their convergence speed. As applications, the split variational inclusion problem, the monotone inclusion problem and the split variational inequality problem are studied. Some numerical experiments are performed to illustrate the computation efficiency of the new algorithms.

Modified Projection Method with Inertial Technique and Hybrid Stepsize for the Split Feasibility Problem

We designed a modified projection method with a new condition of the inertial step and the step size for the split feasibility problem in Hilbert spaces. We show that our iterate weakly converged to a solution. Lastly, we give numerical examples and comparisons that could be applied to signal recovery to show the efficiency of our method.

A hybrid clustering algorithm based on improved GWO and KHM clustering

To solve the problem that the K-means algorithm is sensitive to the initial clustering centers and easily falls into local optima, we propose a new hybrid clustering algorithm called the IGWOKHM algorithm. In this paper, we first propose an improved strategy based on a nonlinear convergence factor, an inertial step size, and a dynamic weight to improve the search ability of the traditional grey wolf optimization (GWO) algorithm. Then, the improved GWO (IGWO) algorithm and the K-harmonic means (KHM) algorithm are fused to solve the clustering problem. This fusion clustering algorithm is called IGWOKHM, and it combines the global search ability of IGWO with the local fast optimization ability of KHM to both solve the problem of the K-means algorithm’s sensitivity to the initial clustering centers and address the shortcomings of KHM. The experimental results on 8 test functions and 4 University of California Irvine (UCI) datasets show that the IGWO algorithm greatly improves the efficiency of the model while ensuring the stability of the algorithm. The fusion clustering algorithm can effectively overcome the inadequacies of the K-means algorithm and has a good global optimization ability.

Subgradient Extragradient Method with Double Inertial Steps for Variational Inequalities

Subgradient Extragradient Method with Double Inertial Steps for Variational Inequalities

Inexact generalized proximal point algorithm with alternating inertial steps for monotone inclusion problem

In this paper, we propose and study an inexact generalized proximal point algorithm with alternated inertial steps for solving monotone inclusion problem and obtain weak convergence results under some mild conditions. In the case when the operator T is such that T-1 is Lipschitz continuous at 0, we prove that the sequence of the iterates is linearly convergent. Fejer monotonicity of even subsequences of the iterates is also obtained. Finally, we give some priori and posteriori error estimates of our generated sequences.

Strongly convergent inertial extragradient type methods for equilibrium problems

This paper studies modified extragradient methods with inertial extrapolation step and self-adaptive step-sizes for solving equilibrium problems in real Hilbert spaces. Strong convergence results are obtained under the assumption that the bifunction is pseudomonotone and satisfies the Lipchitz-type condition. Our method of proof is of independent interest and different from the recent arguments used in related papers on strong convergence methods with inertial steps for equilibrium problems. Numerical implementations and comparisons are given to support the theoretical findings.

The inertial relaxed algorithm with Armijo-type line search for solving multiple-sets split feasibility problem

The multiple-sets split feasibility problem is the generalization of split feasibility problem, which has been widely used in fuzzy image reconstruction and sparse signal processing systems. In this paper, we present an inertial relaxed algorithm to solve the multiple-sets split feasibility problem by using an alternating inertial step. The advantage of this algorithm is that the choice of stepsize is determined by Armijo-type line search, which avoids calculating the norms of operators. The weak convergence of the sequence obtained by our algorithm is proved under mild conditions. In addition, the numerical experiments are given to verify the convergence and validity of the algorithm.

New projection methods with inertial steps for variational inequalities

The inertial projection and contraction method and the inertial forward-backward-forward method have been studied for solving variational inequality problems due to the ability of these methods to work well with only one projection per iteration during computations. The inertial factor in these methods is chosen to be less than 1 in most of the papers that studied these methods. It is known from a computational point of view that the efficiency of these methods improves as the inertial factor approaches 1. In this paper, we modify both the inertial projection and contraction method and the inertial forward-backward-forward method with the possibility of inertial factor taken as 1. Our proposed methods also involve a step-size rule which does not depend on the Lipschitz constant of the cost function and without any line search rule. We analyse the weak convergence of the methods under appropriate conditions. We also modify the proposed methods so that strong convergence is obtained. Preliminary computational results show that our proposed methods are promising and show better performance than some variants of inertial projection and contraction methods and inertial forward-backward-forward methods where the inertial factor is assumed to be less than 1.

Convergence analysis of new inertial method for the split common null point problem

ABSTRACT 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.

Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity

In solving the split variational inequality problems, very few methods have been considered in the literature and most of these few methods require the underlying operators to be co-coercive. This restrictive co-coercive assumption has been dispensed with in some methods, many of which require a product space formulation of the problem. However, it has been discovered that this product space formulation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting structure of the split variational inequality problem. In this paper, we present two new methods with inertial steps for solving the split variational inequality problems in real Hilbert spaces without any product space formulation. We prove that the sequence generated by these methods converges strongly to a minimum-norm solution of the problem when the operators are pseudomonotone and Lipschitz continuous. Also, we provide several numerical experiments of the proposed methods in comparison with other related methods in the literature.

Inertial Derivative-Free Projection Method for Nonlinear Monotone Operator Equations With Convex Constraints

In this paper, we propose an inertial derivative-free projection method for solving convex constrained nonlinear monotone operator equations (CNME). The method incorporates the inertial step with an existing method called derivative-free projection (DFPI) method for solving CNME. The reason is to improve the convergence speed of DFPI as it has been shown and reported in several works that indeed the inertial step can speed up convergence. The global convergence of the proposed method is proved under some mild assumptions. Finally, numerical results reported clearly show that the proposed method is more efficient than the DFPI.

Weak and linear convergence of a generalized proximal point algorithm with alternating inertial steps for a monotone inclusion problem

Weak and linear convergence of a generalized proximal point algorithm with alternating inertial steps for a monotone inclusion problem

Two Hybrid Spectral Methods With Inertial Effect for Solving System of Nonlinear Monotone Equations With Application in Robotics

In this work, motivated by the effect of inertial step in accelerating algorithms for solving nonlinear problems such as equilibrium problems, we propose two hybrid spectral algorithms with inertial effect for solving system of nonlinear equations with convex constraints. The search directions in these algorithms use the convex combination of the modified Barzilai and Borwein spectral parameters (IMA journal of numerical analysis, vol. 8, no. 1, pp. 141-148, 1988) and their geometric mean proposed by Dai et al. (In Numerical Analysis and Optimization, pp. 59-75, Springer, 2015). The incorporation of the inertial-step aids the proposed algorithms in producing more efficient results in comparison with three existing spectral algorithms. Under the assumption that the function under consideration is monotone and satisfies Lipschitz continuity, we prove the global convergence of the proposed algorithms. In addition, we also show the application of the proposed algorithms in motion control of two-joint planar robotic manipulator.

Inertial proximal gradient methods with Bregman regularization for a class of nonconvex optimization problems

This paper proposes an inertial Bregman proximal gradient method for minimizing the sum of two possibly nonconvex functions. This method includes two different inertial steps and adopts the Bregman regularization in solving the subproblem. Under some general parameter constraints, we prove the subsequential convergence that each generated sequence converges to the stationary point of the considered problem. To overcome the parameter constraints, we further propose a nonmonotone line search strategy to make the parameter selections more flexible. The subsequential convergence of the proposed method with line search is established. When the line search is monotone, we prove the stronger global convergence and linear convergence rate under Kurdyka–Łojasiewicz framework. Moreover, numerical results on SCAD and MCP nonconvex penalty problems are reported to demonstrate the effectiveness and superiority of the proposed methods and line search strategy.

Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence

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.

New inertial relaxed method for solving split feasibilities

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.

Inertial Extra-Gradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem

In this paper, we propose a new method, which is set up by incorporating an inertial step with the extragradient method for solving a strongly pseudomonotone equilibrium problems. This method had to comply with a strongly pseudomonotone property and a certain Lipschitz-type condition of a bifunction. A strong convergence result is provided under some mild conditions, and an iterative sequence is accomplished without previous knowledge of the Lipschitz-type constants of a cost bifunction. A sufficient explanation is that the method operates with a slow-moving stepsize sequence that converges to zero and non-summable. For numerical explanations, we analyze a well-known equilibrium model to support our well-established convergence result, and we can see that the proposed method seems to have a significant consistent improvement over the performance of the existing methods.

Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms

This paper derives new inexact variants of the Douglas-Rachford splitting method for maximal monotone operators and the alternating direction method of multipliers (ADMM) for convex optimization. The analysis is based on a new inexact version of the proximal point algorithm that includes both an inertial step and overrelaxation. We apply our new inexact ADMM method to LASSO and logistic regression problems and obtain somewhat better computational performance than earlier inexact ADMM methods.

Projection-type methods with alternating inertial steps for solving multivalued variational inequalities beyond monotonicity

Projection-type methods with alternating inertial steps for solving multivalued variational inequalities beyond monotonicity