极卷积

摘要：

Moreau包络是凸分析中保凸函数的一个重要运算，它是凸优化的发展和分析的核心。本文发展了一种类似的卷积运算的理论，称为极包络，专门用于度规函数。Moreau包络和邻近映射的许多重要性质被极包络及其相应的邻近映射所反映。这些性质包括包络函数的光滑性，邻近映射的唯一性和连续性，它们在对偶性和规范优化算法的构造中起着重要的作用。因此，建立了一套工具来构建这类最优化问题的算法。

英文摘要：

The Moreau envelope is one of the key convexity-preserving functional operations in convex analysis, and it is central to the development and analysis of many approaches for convex optimization. This paper develops the theory for an analogous convolution operation, called the polar envelope, specialized to gauge functions. Many important properties of the Moreau envelope and the proximal map are mirrored by the polar envelope and its corresponding proximal map. These properties include smoothness of the envelope function, uniqueness and continuity of the proximal map, which play important roles in duality and in the construction of algorithms for gauge optimization. A suite of tools with which to build algorithms for this family of optimization problems is thus established.

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