This paper is devoted to the study of sensitivity to perturbation of parametrized variational inclusions involving maximally monotone operators in a Hilbert space. The perturbation of all the data involved in the problem is taken into account. Using the concept of proto-differentiability of a multifunction and the notion of semi-differentiability of a single-valued map, we establish the differentiability of the solution of a parametrized monotone inclusion. We also give an exact formula of the proto-derivative of the resolvent operator associated to the maximally monotone parameterized variational inclusion. This shows that the derivative of the solution of the parametrized variational inclusion obeys the same pattern by being itself a solution of a variational inclusion involving the semi-derivative and the proto-derivative of the associated maps. An application to the study of the sensitivity analysis of a parametrized primal-dual composite monotone inclusion is given. Under some sufficient conditions on the data, it is shown that the primal and the dual solutions are differentiable and their derivatives belong to the derivative of the associated Kuhn–Tucker set.

中文翻译：

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通过解析算子的原始可微性对最大单调内含物进行灵敏度分析

本文致力于研究希尔伯特空间中涉及最大单调算子的参数化变分夹杂物对扰动的敏感性。考虑到问题中涉及的所有数据的扰动。使用多功能的原始可微性概念和单值映射的半可微性概念，我们建立了参数化单调包含解的可微性。我们还给出了与最大单调参数化变分包含相关联的解析算子的原始导数的精确公式。这表明参数化变分包含的解的导数遵循相同的模式，因为它本身是涉及相关映射的半导数和原导数的变分包含的解。给出了在参数化原始-对偶复合单调夹杂物灵敏度分析研究中的应用。在数据的一些充分条件下，表明原解和对偶解是可微的，它们的导数属于相关库恩-塔克集的导数。