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Zero duality gap conditions via abstract convexity
Optimization ( IF 2.2 ) Pub Date : 2021-04-10 , DOI: 10.1080/02331934.2021.1910694
Hoa T. Bui 1 , Regina S. Burachik 2 , Alexander Y. Kruger 3 , David T. Yost 3
Affiliation  

Using tools provided by the theory of abstract convexity, we extend conditions for zero duality gap to the context of non-convex and nonsmooth optimization. Mimicking the classical setting, an abstract convex function is the upper envelope of a family of abstract affine functions (being conventional vertical translations of the abstract linear functions). We establish new conditions for zero duality gap under no topological assumptions on the space of abstract linear functions. In particular, we prove that the zero duality gap property can be fully characterized in terms of an inclusion involving (abstract) ϵ-subdifferentials. This result is new even for the classical convex setting. Endowing the space of abstract linear functions with the topology of pointwise convergence, we extend several fundamental facts of functional/convex analysis. This includes (i) the classical Banach–Alaoglu–Bourbaki theorem (ii) the subdifferential sum rule, and (iii) a constraint qualification for zero duality gap which extends a fact established by Borwein, Burachik and Yao (2014) for the conventional convex case. As an application, we show with a specific example how our results can be exploited to show zero duality for a family of non-convex, non-differentiable problems.



中文翻译:

通过抽象凸性实现零对偶间隙条件

使用抽象凸性理论提供的工具,我们将零对偶间隙的条件扩展到非凸和非光滑优化的上下文中。模仿经典设置,抽象凸函数是抽象仿射函数族的上包络(是抽象线性函数的常规垂直平移)。我们在抽象线性函数空间没有拓扑假设的情况下为零对偶间隙建立了新条件。特别是,我们证明了零对偶间隙属性可以完全用包含(抽象)的包含来表征ε- 次微分。即使对于经典的凸设置,这个结果也是新的。为抽象线性函数的空间赋予逐点收敛的拓扑,我们扩展了函数/凸分析的几个基本事实。这包括 (i) 经典 Banach-Alaoglu-Bourbaki 定理 (ii) 次微分和规则,以及 (iii) 零对偶间隙的约束条件,该条件扩展了 Borwein、Burachik 和 Yao (2014) 为常规凸案子。作为一个应用程序,我们用一个具体的例子展示了如何利用我们的结果来展示一系列非凸、不可微问题的零对偶性。

更新日期:2021-04-10
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