On the strong convergence of a perturbed algorithm to the unique solution of a variational inequality problem

2021/02/11 by Ramzi May, May, Ramzi, Zahrah Bin Ali +1
Computer Science · Mathematics · #47H05 #47H06 #47H09 #Advanced Optimization Algorithms Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Inequalities and Applications #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2102.05861

openalex publication_date 2021/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Q be a nonempty closed and convex subset of a real Hilbert space % H. T:Q→ Q is a nonexpansive mapping which has a least one fixed point. f:Q→ H is a Lipschitzian function, and % F:Q→ H is a Lipschitzian and strongly monotone mapping. We prove, under appropriate conditions on the functions f and F, the control real sequences \α_n\ and \β_n\, and the error term \e_n\, that for any starting point x₀ in Q, the sequence % \x_n\ generated by the perturbed iterative process x_n+1=β_nx_n+(1-β_n)P_Q( α_nf(x_n)+(I-α_nF)Tx_n+e_n) converges strongly to the unique solution of the variational inequality problem Find q∈ C such that ⟨ F(q)-f(q),x-q⟩ ≥ 0 for all x∈ C where C=F_ix(T) is the set of fixed points of T. Our main result unifies and extends many well-known previous results.

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