In this paper, we revisit the numerical approach to some classical variational inequalities, with monotone and Lipschitz continuous mapping A, by means of a projected reflected gradient-type method. A main feature of the method is that it formally requires only one projection step onto the feasible set and one evaluation of the involved mapping per iteration. Contrary to what was done so far, we establish the convergence of the method in a more general setting that allows us to use varying step-sizes without any requirement of additional projections. A linear convergence rate is obtained, when A is assumed to be strongly monotone. Preliminary numerical experiments are also performed.

One-step projected gradient methods for variational inequalities

Abstract In this paper, we study the problem of finding a common solution of the pseudomonotone variational inequality problem and fixed point problem for demicontractive mappings. We introduce a new inertial iterative scheme that combines Tseng’s extragradient method with the viscosity method together with the adaptive step size technique for finding a common solution of the investigated problem. We prove a strong convergence result for our proposed algorithm under mild conditions and without prior knowledge of the Lipschitz constant of the pseudomonotone operator in Hilbert spaces. Finally, we present some numerical experiments to show the efficiency of our method in comparison with some of the existing methods in the literature.

/pdf/strong-convergence-of-a-self-adaptive-inertial-tseng-s-n8b6wrcp.pdf

Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems

In this paper, four self-adaptive iterative algorithms with inertial effects are introduced to solve a split variational inclusion problem in real Hilbert spaces. One of the advantages of the suggested algorithms is that they can work without knowing the prior information of the operator norm. Strong convergence theorems of these algorithms are established under mild and standard assumptions. As applications, the split feasibility problem and the split minimization problem in real Hilbert spaces are studied. Finally, several preliminary numerical experiments as well as an example in the field of compressed sensing are proposed to support the advantages and efficiency of the suggested methods over some existing ones.

Strong Convergence of Self-adaptive Inertial Algorithms for Solving Split Variational Inclusion Problems with Applications

Strong convergence results for quasimonotone variational inequalities

A new projection and contraction method for solving split monotone variational inclusion, pseudomonotone variational inequality, and common fixed point problems

In this work, we investigate pseudomonotone variational inequality problems in a real Hilbert space and propose two projection-type methods with inertial terms for solving them. The first method does not require prior knowledge of the Lipschitz constant and the second one does not require the Lipschitz continuity of the mapping which governs the variational inequality. A weak convergence theorem for our first algorithm is established under pseudomonotonicity and Lipschitz continuity assumptions, and a weak convergence theorem for our second algorithm is proved under pseudomonotonicity and uniform continuity assumptions. We also establish a nonasymptotic O(1/n) convergence rate for our proposed methods. In order to illustrate the computational effectiveness of our algorithms, some numerical examples are also provided.

Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space

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...

A new self-adaptive algorithm for solving pseudomonotone variational inequality problems in Hilbert spaces

Strong Convergence of an Inertial Extragradient Method with an Adaptive Nondecreasing Step Size for Solving Variational Inequalities

A new modified forward–backward–forward algorithm for solving inclusion problems

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Papers

Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space

A new self-adaptive algorithm for solving pseudomonotone variational inequality problems in Hilbert spaces

Strong Convergence of an Inertial Extragradient Method with an Adaptive Nondecreasing Step Size for Solving Variational Inequalities

A new modified forward–backward–forward algorithm for solving inclusion problems

Strong Convergence of an Inertial Extragradient Method with an Adaptive Nondecreasing Step Size for Solving Variational Inequalities