# Qiao-Li DongCivil Aviation University of China | CAU CAFU · College of Science

Qiao-Li Dong

Professor

Optimization algorithms; Fixed point theory and algorithms

## About

131

Publications

19,081

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2,274

Citations

Introduction

My current research interests: iterative algorithm for optimization problem; the fast algorithms for the fixed point of the nonexpansive mappings.

Additional affiliations

July 2008 - April 2017

## Publications

Publications (131)

In this paper, by regarding the two-subspace Kaczmarz method as an alternated inertial randomized Kaczmarz algorithm we present a better convergence rate estimate under a mild condition. Furthermore, we accelerate the alternated inertial randomized Kaczmarz algorithm and introduce a multi-step inertial randomized Kaczmarz algorithm which is proved...

The aim of this paper is to investigate an accelerated Tseng’s extragradient method with double projection for solving Lipschitzian and pseudomonotone variational inequalities in real Hilbert spaces. A strong convergence theorem of the proposed algorithm is obtained under some appropriate assumptions imposed on the parameters. Finally, we give some...

The Korpelevich method is an algorithm which is used to find solutions to equilibrium problems. These problems are mathematical models which are used in economics, game theory, and engineering. Pseudomonotone equilibrium problems are a specific class of equilibrium problems that involve a weakened form of monotonicity. This work introduces a novel...

In this article, we introduce linearized Douglas–Rachford method for solving Lipschitz continuous variational inequalities in Hilbert space. First, we show the linear convergence of linearized Douglas–Rachford method with the fixed stepsize for the strongly monotone mapping. The usual drawback of algorithms with the fixed stepsize is the requiremen...

In this paper, we propose a new algorithm with inertial term and self-adaptive stepsize for solving the split variational inclusion problem (denoted by SVIP) in real Hilbert spaces. Under suitable conditions imposed on the parameters, we prove that our iterative scheme converges strongly to an element of the solution set of SVIP without the prior k...

This article is concerned with a universal version of projected reflected gradient method with new step size for solving variational inequality problem in Hilbert spaces. Under appropriate assumptions controlled by the operators and parameters, we acquire the weak convergence of the proposed algorithm. Moreover, we establish an R-linear convergence...

In this paper, by regarding the two-subspace Kaczmarz method [20] as an alternated inertial randomized Kaczmarz algorithm we present a new convergence rate estimate which is shown to be better than that in [20] under a mild condition. Furthermore, we accelerate the alternated inertial randomized Kaczmarz algorithm and introduce a multi-step inertia...

In this paper, we introduce a three-operator splitting algorithm with deviations for solving the minimization problem composed of the sum of two convex functions minus a convex and smooth function in a real Hilbert space. The main feature of the proposed method is that two per-iteration deviation vectors provide additional degrees of freedom. We pr...

We propose an adaptive way to choose the anchoring parameters for the Halpern iteration to find a fixed point of a nonexpansive mapping in a real Hilbert space. We prove strong convergence of this adaptive Halpern iteration and obtain the rate of asymptotic regularity at least O ( 1 / k ) O(1/k) , where k k is the number of iterations. Numerical ex...

In this paper, we introduce a new modified self adaptive subgradient extragradient algorithm for solving pseudomonotone variational inequality problems and fixed point problems in Hilbert spaces. Precisely, we prove that the sequence generated by our algorithm is strongly convergent under some suitable conditions imposed on the parameters. Moreover...

In this paper we propose a new probability distribution for the randomized Kaczmarz (RK) algorithm where each row of the coefficient matrix is selected in the current iteration with the probability proportional to the square of the sine of the angle between it and the chosen row in the previous iteration. This probability distribution is helpful to...

This paper aims to introduce a new projection method for solving pseudomonotone variational inequality problems in real reflexive Banach spaces. The main algorithm is based on the self-adaptive method, subgradient extragradient method and Bregman projection method. Under some appropriate assumptions imposed on the parameters, we prove a strong conv...

Our study in this paper is focused on the split equality fixed-point problem with firmly quasi-non-expansive operators in infinite-dimensional Hilbert spaces. A self-adaptive simultaneous scheme is introduced, and its weak convergence is established under mild and standard assumptions. The new proposed scheme generalizes and extends some related wo...

The aim of this paper is to introduce a new subgradient extragradient algorithm for solving variational inequality problems involving pseudomonotone and uniformly continuous operator in Banach spaces. Moreover, we prove a strong convergence theorem by constructing a new line-search rule. At the same time, several numerical experimental results are...

The projection technique is a very important method and efficient for solving variational inequality problems. In this study, we developed the subgradient extragradient method for solving pseudomonotone variational inequality in real Hilbert spaces. Our first algorithm requires only computing one projection onto the feasible set per iteration and t...

The purpose of this paper is to investigate pseudomonotone variational inequalities in real Hilbert spaces. For solving this problem, we introduce a new method. The proposed algorithm combines the advantages of the subgradient extragradient method and the projection and contraction method. We establish the strong convergence of the proposed algorit...

We study the forward-backward-forward splitting method with alternated inertial step in this paper. We consider under-relaxation and over-relaxation approaches to the proposed method and obtain weak convergence results in both cases in real Hilbert spaces. Linear convergence alongside priori and apriori error estimates are obtained under some stand...

In this work, we propose two new iterative schemes for finding an element of the set of solutions of a pseudo-monotone, Lipschitz continuous variational inequality problem in real Hilbert spaces. The weak and strong convergence theorems are presented. The advantage of the proposed algorithms is that they do not require prior knowledge of the Lipsch...

Motivated and inspired by the works of Ceng et al. (2010) and Yao and Postolache (2012), we first study a relaxed inertial Tseng’s method for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of fixed points of an κ-demicontractive mapping in real Hilbert spaces. The...

In this paper, we propose a new version of the extragradient method for solving non-Lipschitzian and pseudo-monotone variational inequalities in real Hilbert spaces. First, we show that the proposed method converges strongly to the minimum-norm solution of a variational inequality under mild assumptions. Second, we obtain a linear convergence rate...

In this paper, we propose an alternated inertial subgradient extragradient algorithm for variational inequalities with self-adaptive step-sizes and obtain weak and linear convergence results. We also obtain linear convergence results using an alternated inertial projected gradient algorithm for which knowledge of the modulus of strong pseudo-monoto...

In this paper, we propose an algorithm combining Bregman alternating minimization algorithm with two-step inertial force for solving a minimization problem composed of two nonsmooth functions with a smooth one in the absence of convexity. For solving nonconvex and nonsmooth problems, we give an abstract convergence theorem for general descent metho...

In this paper, we propose an alternated inertial general splitting method with linearization for a split feasibility problem. Four rules of inertial parameters and relaxation parameters are discussed, where the adaptive inertial parameters are firstly investigated. The convergence of the proposed method is established under standard conditions. The...

In this paper, we propose a viscosity iterative algorithm with alternated inertial extrapolation step to solve the split feasibility problem, where the self-adaptive stepsize is used. Under appropriate conditions, the proposed algorithm is proved to converge to a solution of the split feasibility problem, which is also the unique solution of a vari...

The aim of this article is to introduce the Douglas–Rachford splitting method with linearization to solve the split feasibility problem (SFP). Our proposed method includes two existing methods in work of Tang et al. and Wang as special cases. The ranges of the parameters in work of Tang et al. are extended from (0,1) to (0,2). Under standard condit...

In this paper, we introduce a numerical iterative algorithm with a reflected step to solve the equilibrium problem, which involves non-monotone bifunctions, in real Hilbert spaces. We give weak convergence analysis when the bifunctions are convex and jointly weakly continuous alongside the associated Minty equilibrium problem with a solution. The a...

It is known that several optimization problems can be converted to a fixed point problem for which the underline fixed point operator is an averaged quasi-nonexpansive mapping and thus the corresponding fixed point method utilizes to solve the considered optimization problem. In this paper, we consider a fixed point method involving inertial extrap...

In this paper, we investigate a new inertial viscosity extragradient algorithm for solving variational inequality problems for pseudo-monotone and Lipschitz continuous operator and fixed point problems for quasi-nonexpansive mappings in real Hilbert spaces. Strong convergence theorems are obtained under some appropriate conditions on the parameters...

In this paper, we introduce a new subgradient extragradient method for solving variational inequalities involving pseudomonotone and Lipschitz operators in 2-uniformly convex and uniformly smooth Banach spaces. The convergence rate of the proposed algorithm is accelerated as we use a new step size rule to control the iterative sequence. Finally, we...

In this chapter, we present some concepts, definitions, and lemmas which will be used in the following chapters.

This chapter discusses two applications of the Krasnosel’skiı̆–Mann iteration: asynchronous parallel coordinate updates methods and cyclic coordinate update algorithms. Their convergence analysis and convergence rate estimate are established under some conditions.

Some operator splitting methods can be treated as particular cases of the Krasnosel’skiı̆–Mann iteration for finding fixed points of nonexpansive operators or averaged operators. Since the Krasnosel’skiı̆–Mann iteration has the advantage of allowing the inclusion of relaxation parameters in the update rules of the iterates, it is often used in the...

In the book [164], Ortega and Rheinboldt introduced a general iterative process:
$$\displaystyle x_{n+1}=\varTheta _n(x_n,x_{n-1},\cdots ,x_{n-s+1})\,\,\,\,\mbox{for each}\,\,\, n\geq 1, $$
where s ≥ 1 is an integer and Θ
n(⋅) is the function that performs “extrapolation” onto the points x
n, x
n−1, ⋯ , x
n−s+1. The iterative process (6.1) is calle...

In this chapter, we discuss the progress of the original Krasnosel’skiı̆–Mann iteration, the perturbations of the Krasnosel’skiı̆–Mann iteration, and several convergence rates.

The inertial type algorithms [172] originate from the heavy ball method of the so-called heavy ball with friction dynamical system:
$$\displaystyle \ddot {x}(t)+\gamma (t) \dot {x}(t)+\nabla \varphi (x(t))=0, $$
where φ is differentiable.

In this chapter, we present some ranges and several optimal choices of some relaxation parameter sequence {λ
n} of the Krasnosel’skiı̆–Mann iteration (3.1) in the theory and actual practice. Some variants of the Krasnosel’skiı̆–Mann iteration are obtained based on the equivalence of the fixed point problem, the variational inequality problem, and n...

This work proposes two different primal-dual splitting algorithms for solving structured monotone inclusion containing a cocoercive operator and the parallel-sum of maximally monotone operators. In particular, the parallel-sum is symmetry. The proposed primal-dual splitting algorithms are derived from two approaches: One is the preconditioned forwa...

The focus of this paper is to obtain weak and linear convergence analysis of the subgradient extragradient method with alternated inertial step for solving equilibrium problems in real Hilbert spaces. The proposed method uses self-adaptive step sizes. Weak convergence is established without Lipschitz constant of the bifunction as an input parameter...

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 t...

The purpose of this paper is to study a new Tseng’s extragradient method with two different stepsize rules for solving pseudomonotone variational inequalities in real Hilbert spaces. We prove a strong convergence theorem of the proposed algorithm under some suitable conditions imposed on the parameters. Moreover, we also give some numerical experim...

In this paper, we introduce a Totally Relaxed Self adaptive Subgradient Extragradient Method (TRSSEM) with Halpern iterative scheme for finding a common solution of Variational Inequality Problem (VIP) and fixed point of quasi-nonexpansive mapping in a 2-uniformly convex and uniformly smooth Banach space. The TRSSEM does not require the computation...

The aim of this paper is to give a strong convergence theorem of a new iterative algorithm for solving variational inequalities with pseudomonotone and non-Lipschitzian operators in real Hilbert spaces. The proposed algorithm combines the inertial method and the extragradient method which simplifies and accelerates the process of convergence by est...

In this article, we introduce several Douglas-Rachford method to solve the split feasibility problems (SFP). Firstly, we propose a new iterative method by combining Douglas-Rachford method and Halpern iteration. The stepsize is determined dynamically which does not need any prior information about the operator norm. A relaxed version is presented f...

Recently, the authors (Dong et al. in J Global Optim 73(4):801–824, 2019) introduced the multi-step inertial Krasnosel’skiǐ–Mann iteration, where the inertial parameters involve the iterative sequence. Therefore, one has to compute the inertial parameters per iteration. The aim of this article is to present two kinds of inertial parameter arrays wh...

The split feasibility problem is to find a point x * with the property that x * ∈ C and Ax * ∈ Q, where C and Q are nonempty closed convex subsets of real Hilbert spaces X and Y , respectively, and A is a bounded linear operator from X to Y. The split feasibility problem models inverse problems arising from phase retrieval problems and the intensit...

The purpose of this article is to introduce a general inertial projected gradient method with a self-adaptive stepsize for solving variational inequality problems. The proposed method incorporates two different extrapolations with respect to the previous iterates into the projected gradient method. The weak convergence for our method is proved unde...

We propose and study new projection-type algorithms for solving pseudomonotone variational inequality problems in real Hilbert spaces without assuming Lipschitz continuity of the cost operators. We prove weak and strong convergence theorems for the sequences generated by these new methods. The numerical behavior of the proposed algorithms when appl...

The aim of this paper is to introduce a new inertial Tseng’s extragradient algorithm for solving variational inequality problems with pseudo-monotone and Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in real Hilbert spaces. We prove a strong convergence theorem for the proposed algorithm under suitable assumptions...

The authors [13] introduced a general inertial Krasnosel’skiǐ–Mann algorithm: yn=xn+αn(xn-xn-1),zn=xn+βn(xn-xn-1),xn+1=(1-λn)yn+λnT(zn)for each n≥1 and showed its convergence with the control conditions αn,βn∈[0,1). In this paper, we present the convergence analysis of the general inertial Krasnosel’skiǐ–Mann algorithm with the control conditions α...

In this paper, we revisit the subgradient extragradient method for solving a pseudomonotone variational inequality problem with the Lipschitz condition in real Hilbert spaces. A new algorithm based on the subgradient extragradient method with the technique of choosing a new step size is proposed. The weak convergence of the proposed algorithm is es...

In this article, we introduce a general splitting method with linearization to solve the split feasibility problem and propose a way of selecting the stepsizes such that the implementation of the method does not need any prior information about the operator norm. We present the constant and adaptive relaxation parameters, and the latter is "optimal...

In this work, we propose a new modified Popov’s method by using inertial effect for solving the variational inequality problem in real Hilbert spaces. The advantage of the proposed algorithm is the computation of only one value of the inequality mapping and one projection onto the admissible set per one iteration as well as it does not need to the...

In this paper, we first study a relaxed inertial projection and contraction method for variational inequalities to obtain weak convergence results under standard assumptions in Hilbert spaces. Next, we propose another inertial projection and contraction method for which the inertial factor θ is chosen in [0,1] with θ=1 possible. Weak and linear con...

It is known that several optimization problems can be converted to a fixed point problem for which the underline fixed point operator is an averaged quasi-nonexpansive mapping and thus the corresponding fixed point method utilize to solve the considered optimization problem. In this paper, we consider a fixed point method involving inertial extrapo...

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 assumption...

The focus of this paper is to introduce algorithms with alternated inertial step to solve split feasibility problems. We obtain global convergence of the sequences of iterates generated by the proposed methods under some appropriate conditions. When the split feasibility problem satisfies some bounded linear regularity property, we show that the ge...

In this paper, we propose a new viscosity extragradient algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz continuous operator in real Hilbert spaces. We prove a strong convergence theorem under some appropriate conditions imposed on the parameters. Finally, we give some numerical experiments to illustrate th...

The purpose of this paper is to study the convergence analysis of an iterative algorithm with inertial extrapolation step for finding an approximate solution of split monotone inclusion problem in real Hilbert spaces. Weak convergence of the sequence of iterates generated from the proposed method is obtained under some mild assumptions. Some specia...

In this paper, we propose a single projection method for finding a solution of the bilevel pseudo-monotone variational inequality problem in real Hilbert spaces. The advantage of the proposed algorithm requires only one projection onto the feasible set. Also, we prove strong convergence theorems of the proposed method under mild conditions, which i...

In this paper, we consider an improvement of the extragradient method to figure out the numerical solution for pseudomonotone equilibrium problems in arbitrary real Hilbert space. A new method is proposed with an inertial scheme and a self adaptive step size rule that is revised on each iteration based on the previous three iterations. The weak con...

In this paper, we proposed two different methods for solving pseudomonotone and strongly pseudomonotone equilibrium problems. We can examine these methods as an extension and improvement of the Popov's extragradient method. We replaced the second minimization problem onto a closed convex set in the Popov's extragradient method, with a half-space mi...

The purpose of this paper is to study a new viscosity iterative algorithm for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping and the set of solutions of a new variational inequality problem involving inverse-strongly monotone operators in Hilbert spaces. We prove some strong convergence theorems under...

The main purpose of this paper is to introduce a modified inertial forward-backward
splitting method and prove its strong convergence to a zero of the sum of two accretive operators in real uniformly convex Banach space which is also uniformly smooth. We then apply our results to solve variational inequality problem and convex minimization problem....

In this work, we propose a new modified Popov’s method by using inertial effect
for solving the variational inequality problem in real Hilbert spaces. The advantage of the
proposed algorithm is the computation of only one value of the inequality mapping and one projection onto the admissible set per one iteration as well as it does not need to the...

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 opera...

This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'skiǐ-Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnosel'skiǐ-Mann iteration, with which the algorithm is proved to achieve the optimal convergence rate. Then we present an approximation to the optimal relax...

In this paper, we consider a special split feasibility problem (SFP): x∈CandAx∈Q where C is the solution set of an equilibrium problem, Q is a convex subset in Rm, and A:Rn→Rm is a linear operator. We introduce two projection algorithms for solving the SFP by combining the projection method for the equilibrium problem and the gradient method for th...

In this paper, we propose two extragradient methods for finding an element of
the set of solutions of the bilevel pseudo-monotone variational inequality problems in real
Hilbert spaces. The advantage of proposed algorithms requires only one projection onto thefeasible set. The strong convergence theorems are proved under mild conditions. Our result...

In this article, we introduce the multi-step inertial proximal contraction algorithms (MiPCA) to approximate a zero of the sum of two monotone operators, with one of the two operators being monotone and Lipschitz continuous. The weak convergence of the MiPCA is shown under the summability condition formulated in terms of the iterative sequence in a...

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 assumption...

In this paper, we suggest two inertial Krasnosel’skiǐ–Mann type hybrid algorithms to approximate a solution of a hierarchical fixed point problem for nonexpansive mappings in Hilbert space. We prove strong convergence theorems for these algorithms and the conditions of the convergence are very weak comparing other algorithms for the hierarchical fi...

In this paper, we first introduce a multi-step inertial Krasnosel’skiǐ–Mann algorithm (MiKM) for nonexpansive operators in real Hilbert spaces. We give the convergence of the MiKM by investigating the convergence of the Krasnosel’skiǐ–Mann algorithm with perturbations. We also establish global pointwise and ergodic iteration complexity bounds of th...