This paper introduces a proximal bundle scheme to solve generalized variational inequalities with inexact data. Under optimality conditions, the problem can be equivalently represented as seeking out the zero point of the sum of two multi-valued operators whose domains are the real Hilbert space. The two operators denoted by T and f, respectively, are the monotone operator and the subdifferential of a lower semi-continuous, non-differentiable, convex function. Our approach is based on the principles of the proximal point strategy, which involves incorporating inexact information into the subproblems and approximating them using a series of piecewise linear convex functions. Moreover, we put forward a novel stopping criterion to identify the adequacy of the current approximation. This approach serves to make the subproblems more manageable, and it has been proven that obtaining inexact information can ensure that the linearization error during the iteration process remains non-negative, thus avoiding triggering noise attenuation. Subsequently, we verify the convergence of the algorithm under relatively mild assumptions (the operator T is para-monotone and may be multi-valued). Ultimately, we present the findings of elementary numerical experiments to declare the method's efficacy.

中文翻译：

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一种用于求解具有不精确数据的广义变分不等式的近端束方法


本文介绍了一种近端束方案来解决具有不精确数据的广义变分不等式。在最优性条件下，该问题可以等效地表示为寻找两个多值运算符之和的零点，这两个运算符的域是真实的希尔伯特空间。用 T 和 f 表示的两个运算符分别是单调运算符和下半连续、不可微分凸函数的子微分。我们的方法基于近端点策略的原理，该策略涉及将不精确的信息合并到子问题中，并使用一系列分段线性凸函数对其进行近似。此外，我们提出了一种新的停止准则来识别电流近似的充分性。这种方法有助于使子问题更易于管理，并且已经证明，获得不准确的信息可以确保迭代过程中的线性化误差保持非负值，从而避免触发噪声衰减。随后，我们在相对温和的假设下验证了算法的收敛性（算子 T 是准单调的，可能是多值的）。最后，我们提出了基本数值实验的结果来宣布该方法的有效性。