Abstract
In this paper, we introduce two novel extragradient-like methods to solve variational inequalities in a real Hilbert space. The variational inequality problem is a general mathematical problem in the sense that it unifies several mathematical models, such as optimization problems, Nash equilibrium models, fixed point problems, and saddle point problems. The designed methods are analogous to the two-step extragradient method that is used to solve variational inequality problems in real Hilbert spaces that have been previously established. The proposed iterative methods use a specific type of step size rule based on local operator information rather than its Lipschitz constant or any other line search procedure. Under mild conditions, such as the Lipschitz continuity and monotonicity of a bi-function (including pseudo-monotonicity), strong convergence results of the described methods are established. Finally, we provide many numerical experiments to demonstrate the performance and superiority of the designed methods.
Highlights
This paper concerns the problem of the classic variational inequality problem [1,2]
We provide a positive answer of this question, i.e., the gradient method still generates a strong convergence sequence by using a fixed and variable step size rule for solving a problem (VIP) associated with pseudo-monotone mappings
Main Results we introduce both inertial-type subgradient extragradient methods which incorporate a monotone step size rule and the inertial term and provide both strong convergence theorems
Summary
This paper concerns the problem of the classic variational inequality problem [1,2]. The variational inequalities problem (VIP) for an operator G : H → H is defined in the following way: Find u∗ ∈ C such that G(u∗), y − u∗ ≥ 0, ∀ y ∈ C (VIP)where C is a non-empty, convex and closed subset of a real Hilbert space H and ., . and . denote an inner product and the induced norm on H, respectively. A natural question arises: “Is it possible to introduce a new inertial-type strongly convergent extragradient-like method with a monotone variable step size rule to solve problem (VIP)”? We provide a positive answer of this question, i.e., the gradient method still generates a strong convergence sequence by using a fixed and variable step size rule for solving a problem (VIP) associated with pseudo-monotone mappings.
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