Abstract
The objective of this article is to solve pseudomonotone variational inequality problems in a real Hilbert space. We introduce an inertial algorithm with a new self-adaptive step size rule, which is based on the projection and contraction method. Only one step projection is used to design the proposed algorithm, and the strong convergence of the iterative sequence is obtained under some appropriate conditions. The main advantage of the algorithm is that the proof of convergence of the algorithm is implemented without the prior knowledge of the Lipschitz constant of cost operator. Numerical experiments are also put forward to support the analysis of the theorem and provide comparisons with related algorithms.
Highlights
Let H be a real Hilbert space with the scalar product ·, · and the induced norm ·
In this paper, motivated and inspired by the results in the literature Tseng [7], Thong and Hieu [11, 12], and by the ongoing research in these directions, we introduce a new algorithm for solving the (VIP) involving pseudomonotone and Lipschitz continuous operator
The algorithm combines the inertial technique with the projection and contraction method (PCM), it uses a new step size rule which allows the introduced algorithm to work without depending on the Lipschitz constant of cost operator, the step size is updated over each iteration
Summary
Let H be a real Hilbert space with the scalar product ·, · and the induced norm ·. The method (PCM) requires only one step projection onto the feasible set in each iteration, and the sequence {xn} generated by the PCM converges weakly to a point in VI( , A) under suitable conditions. Note that only one step projection is required by the algorithm, and the strong convergence theorem is proved This algorithm was studied with a self-adaptive technique so that the conditions imposed on the cost operator can be relaxed. The algorithm combines the inertial technique with the projection and contraction method (PCM), it uses a new step size rule which allows the introduced algorithm to work without depending on the Lipschitz constant of cost operator, the step size is updated over each iteration. Under several appropriate conditions on the parameters, we will prove that the sequence {xn} generated by the new algorithm converges strongly to a minimum-norm solution.
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