Basic Convex Analysis in Metric Spaces with Bounded Curvature
Basic Convex Analysis in Metric Spaces with Bounded Curvature
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DOI：
10.1137/23m1551389
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发表时间：
2023-02
期刊：
影响因子：
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通讯作者：
A. Lewis;Genaro L'opez-Acedo;A. Nicolae
中科院分区：
文献类型：
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作者：
A. Lewis;Genaro L'opez-Acedo;A. Nicolae
可微结构确保了许多经典凸分析的基础自然地从欧几里得空间扩展到黎曼流形。但是，如果没有这样的结构，扩展将更具挑战性。尽管如此，在曲率有界（但可能是正的）的亚历山德罗夫空间中，我们开发了几个基本的构建块。我们通过投影和法锥定义了次梯度，证明了它们的存在性，并将它们与经典仿射微小性质联系起来。然后，在相当于简单微积分或对偶结果的情况下，我们开发了最小化两个凸函数和的必要最优性条件。
Differentiable structure ensures that many of the basics of classical convex analysis extend naturally from Euclidean space to Riemannian manifolds. Without such structure, however, extensions are more challenging. Nonetheless, in Alexandrov spaces with curvature bounded above (but possibly positive), we develop several basic building blocks. We define subgradients via projection and the normal cone, prove their existence, and relate them to the classical affine minorant property. Then, in what amounts to a simple calculus or duality result, we develop a necessary optimality condition for minimizing the sum of two convex functions.