In this paper we associate with an infinite family of real extended functions defined on a locally convex space a sum, called robust sum, which is always well-defined. We also associate with that family of functions a dual pair of problems formed by the unconstrained minimization of its robust sum and the so-called optimistic dual. For such a dual pair, we characterize weak duality, zero duality gap, and strong duality, and their corresponding stable versions, in terms of multifunctions associated with the given family of functions and a given approximation parameter ε ≥ 0 which is related to the ε-subdifferential of the robust sum of the family. We also consider the particular case when all functions of the family are convex, assumption allowing to characterize the duality properties in terms of closedness conditions.

中文翻译：

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功能强大的和的对偶

在本文中，我们将无限个实扩展函数族定义在局部凸空间上的总和称为鲁棒总和，该总和始终是定义明确的。我们还将与该功能族相关联的是由其鲁棒和不受限制地最小化而形成的双对问题，即所谓的乐观对偶。对于这样的双对，我们表征弱对偶，零偶间隙，和强对偶性，和它们的相应的稳定版本，在与功能的给定家族和相关联的多值映射的术语给定的近似参数ε ≥0，这是关系到ε-家庭稳健和的亚微分。我们还考虑了该族的所有功能都是凸的情况，这种假设允许根据封闭性条件刻画二元性。