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<p>We consider the problem of solving pseudomonotone variational inequality problems (PVIPs) which are subject to constraints arising from a monotone variational inclusion problem (MVIP) and a common fixed-point problem (CFPP). The CFPP involves a relatively asymptotically nonexpansive operator together with a finite family of relatively nonexpansive operators. By invoking Tseng’s extragradient method with a proximal–contraction strategy, we propose and analyze an adaptive inertial subgradient-like extragradient algorithm for solving a pair of PVIPs under MVIP and CFPP constraints in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>-uniformly convex and uniformly smooth real Banach space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(E\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>E</mi> </math></EquationSource> </InlineEquation>. Under mild assumptions, we establish weak convergence results for the proposed algorithms. Notably, unlike several existing methods in the literature, the proposed algorithms do not require prior knowledge of the Lipschitz constants of the underlying operators. Finally, the application to image restoration and some numerical examples are presented to demonstrate the effectiveness and practical feasibility of the proposed algorithms.</p>

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On Inertial Subgradient-like Extragradient Methods with Proximal Contraction Strategy for Variational Inequalities

  • 期刊论文

  • 发表日期：

    2026年6月17日

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  • Lu-Chuan Ceng,
  • Ajay Kumar,
  • Adrian Petruşel,
  • Xiaopeng Zhao

摘要

We consider the problem of solving pseudomonotone variational inequality problems (PVIPs) which are subject to constraints arising from a monotone variational inclusion problem (MVIP) and a common fixed-point problem (CFPP). The CFPP involves a relatively asymptotically nonexpansive operator together with a finite family of relatively nonexpansive operators. By invoking Tseng’s extragradient method with a proximal–contraction strategy, we propose and analyze an adaptive inertial subgradient-like extragradient algorithm for solving a pair of PVIPs under MVIP and CFPP constraints in \(2\) 2 -uniformly convex and uniformly smooth real Banach space \(E\) E . Under mild assumptions, we establish weak convergence results for the proposed algorithms. Notably, unlike several existing methods in the literature, the proposed algorithms do not require prior knowledge of the Lipschitz constants of the underlying operators. Finally, the application to image restoration and some numerical examples are presented to demonstrate the effectiveness and practical feasibility of the proposed algorithms.