SINGLE PROJECTION ALGORITHM FOR VARIATIONAL INEQUALITIES IN BANACH SPACES WITH APPLICATION TO CONTACT PROBLEM

doi:10.1007/s10473-020-0412-2

摘要/Abstract

摘要： We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern's iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function. Finally, we give an example of a contact problem where our proposed method can be applied.

关键词: variational inequality, 2-uniformly convex Banach space, Tseng's algorithm, strong convergence, rate of convergence

Abstract: We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern's iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function. Finally, we give an example of a contact problem where our proposed method can be applied.

Key words: variational inequality, 2-uniformly convex Banach space, Tseng's algorithm, strong convergence, rate of convergence

中图分类号:

47H05

Yekini SHEHU. SINGLE PROJECTION ALGORITHM FOR VARIATIONAL INEQUALITIES IN BANACH SPACES WITH APPLICATION TO CONTACT PROBLEM[J]. 数学物理学报(英文版), 2020, 40(4): 1045-1063.

Yekini SHEHU. SINGLE PROJECTION ALGORITHM FOR VARIATIONAL INEQUALITIES IN BANACH SPACES WITH APPLICATION TO CONTACT PROBLEM[J]. Acta mathematica scientia,Series B, 2020, 40(4): 1045-1063.

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