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.

中文翻译：

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Banach空间变分不等式的单投影算法在接触问题中的应用

我们研究了 Tseng 的单投影算法，用于解决 2-一致凸 Banach 空间中的变分不等式问题。假设变分不等式的下划线成本函数是单调的且 Lipschitz 连续的。在对可变步长的合理假设下获得弱收敛结果。我们还给出了下划线成本函数是强单调和 Lipchitz 连续时的强收敛结果。对于这种强收敛情况，所提出的方法不需要强单调性模数和成本函数的 Lipschitz 常数的先验知识作为输入参数，相反，可变步长是递减且不可求和的。还给出了强收敛情况下收敛速度的渐近估计。为了完整性，当成本函数是单调和 Lipschitz 连续的并且可变步长以成本函数的 Lipschitz 常数的倒数为界时，我们使用 Halpern 迭代的思想给出了另一个强收敛结果。最后，我们给出了一个可以应用我们提出的方法的接触问题的例子。