Abstract
In a Hadamard manifold, let the VIP and SVI represent a variational inequality problem and a system of variational inequalities, respectively, where the SVI consists of two variational inequalities which are of symmetric structure mutually. This article designs two parallel algorithms to solve the SVI via the subgradient extragradient approach, where each algorithm consists of two parts which are of symmetric structure mutually. It is proven that, if the underlying vector fields are of monotonicity, then the sequences constructed by these algorithms converge to a solution of the SVI. We also discuss applications of these algorithms for approximating solutions to the VIP. Our theorems complement some recent and important ones in the literature.
Highlights
Suppose that the operator F is a self-mapping on a real Hilbert space (H, h·, ·i)
It is well known that variational inequalities like variational inequality problem (VIP) (1) have played an important role in the study of economics, transportation, mathematical programming, engineering mechanics, etc
It is clear that the system of variational inequalities (SVI) (3) consists of two variational inequalities which are of symmetric structure mutually
Summary
Suppose that the operator F is a self-mapping on a real Hilbert space (H, h·, ·i). Let the set C ⊂ H be nonempty, convex, and closed. In 1976, Korpelevich’s extragradient rule was first introduced in [1] for solving VIP (1). According to problems (3) and (4), Ceng et al [22] introduced the new SVI of finding (u∗ , v∗ ) ∈ C × C s.t. where constants μ1 , μ2 ∈ (0, ∞), and exp−1 is the inverse of exponential map. We design two parallel algorithms to solve the SVI (5) via the subgradient extragradient approach, where each algorithm consists of two parts which have a mutually symmetric structure. If the underlying vector fields are of monotonicity, the sequences constructed by these algorithms converge to a solution of the SVI (5).
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