<p>In almost all existing projection and contraction methods including their modifications, the range of the constant <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> is (0, 2) and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ \{\rho _n\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>ρ</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> has the similar definitions. In this paper, we introduce a new inertial subgradient projection and contraction method for solving a variational inequality problem in Hilbert space. In our method, the mapping is not required to be pseudomonotone, the range of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> is relaxed from (0, 2) to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ (0,\infty ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ \{\rho _n\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>ρ</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is computed by a new manner and the self-adaptive step size admitted to be increasing is used for dealing with the unknown Lipschitz constant. Under some new conditions, we prove the strong convergence of the proposed method. Some numerical examples and an application are presented to illustrate the effectiveness of our method and compare the numerical results with some related methods in the literature.</p>

Modified inertial subgradient projection and contraction method for solving nonmonotone variational inequality problem in Hilbert space

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发表日期：

2025年6月5日

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Shenghua Wang,
Yueyao Zhang,
Yeo Je Cho

摘要

In almost all existing projection and contraction methods including their modifications, the range of the constant $ \gamma $ $γ$ is (0, 2) and $ \{\rho _n\} $ ${ρ_{n}}$ has the similar definitions. In this paper, we introduce a new inertial subgradient projection and contraction method for solving a variational inequality problem in Hilbert space. In our method, the mapping is not required to be pseudomonotone, the range of $ \gamma $ $γ$ is relaxed from (0, 2) to $ (0,\infty ) $ $(0, \infty)$ , $ \{\rho _n\} $ ${ρ_{n}}$ is computed by a new manner and the self-adaptive step size admitted to be increasing is used for dealing with the unknown Lipschitz constant. Under some new conditions, we prove the strong convergence of the proposed method. Some numerical examples and an application are presented to illustrate the effectiveness of our method and compare the numerical results with some related methods in the literature.