Abstract
The aim of this article is to study two efficient parallel algorithms for obtaining a solution to a system of monotone variational inequalities (SVI) on Hadamard manifolds. The parallel algorithms are inspired by Tseng’s extragradient techniques with new step sizes, which are established without the knowledge of the Lipschitz constants of the operators and line-search. Under the monotonicity assumptions regarding the underlying vector fields, one proves that the sequences generated by the methods converge to a solution of the monotone SVI whenever it exists.
Highlights
Given an operator A : H → H and a convex and closed subset C in a real Hilbert space H, the well known variational inequality problem (VIP) indicates the one of finding a point x ∗ ∈ C such that h Ax ∗, x − x ∗ i ≥ 0 ∀ x ∈ C. (1)It is well known that the variational inequality theory has been playing a big role in the study of signal processing, image reconstruction, mathematical programming, differential equations, and others; see, e.g., [1,2,3,4,5]
Inspired by problems (3) and (4), this paper introduces and considers the system of monotone variational inequalities (SVI) on Hadamard manifolds, that is, find ( x ∗, y∗ ) ∈ C × C such that y∗ x + μ1 A1 y, expx ∗ x i ≥ 0 ∀ x ∈ C, 1 ∗
A1 = A2 = A in Algorithms 1 and 3, these algorithms are reduced to the following Algorithms 2 and 4, respectively, for solving the monotone VIP (4) on Hadamard manifolds
Summary
Given an operator A : H → H and a convex and closed subset C in a real Hilbert space H, the well known variational inequality problem (VIP) indicates the one of finding a point x ∗ ∈ C such that h Ax ∗ , x − x ∗ i ≥ 0 ∀ x ∈ C. (1)It is well known that the variational inequality theory has been playing a big role in the study of signal processing, image reconstruction, mathematical programming, differential equations, and others; see, e.g., [1,2,3,4,5]. Given an operator A : H → H and a convex and closed subset C in a real Hilbert space H, the well known variational inequality problem (VIP) indicates the one of finding a point x ∗ ∈ C such that h Ax ∗ , x − x ∗ i ≥ 0 ∀ x ∈ C. A number of investigators have conducted various approaches on algorithms; see e.g., [12,13,14,15,16,17,18] and the references cited therein. Let both A1 and A2 be single-valued self-operators on space H. Ceng et al [19] considered and studied the following system problem of finding (x∗ , y∗ ) ∈ C × C such that h x ∗ − y∗ + μ1 A1 y∗ , x − x ∗ i ≥ 0 ∀ x ∈ C, hy∗ − x ∗ + μ2 A2 x ∗ , x − y∗ i ≥ 0 ∀ x ∈ C, Symmetry 2020, 12, 43; doi:10.3390/sym12010043
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