FOUNDATIONS OF COMPUTATIONALMATHEMATICS Self-Scaled Barrier Functions on Symmetric Cones and Their Classification
FOUNDATIONS OF COMPUTATIONALMATHEMATICS Self-Scaled Barrier Functions on Symmetric Cones and Their Classification
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计算数学基础 对称锥上的自标度势垒函数及其分类
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通讯作者：
Raphael Andreas Hauser
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作者：
Raphael Andreas Hauser
。自缩放锥体上的自缩放势垒函数由 Nesterov 和 Todd 于 1994 年公理化地引入，作为构建原始对偶长步内点算法的工具。本文通过展示这些对象的对称性、它们与其定义域的对称群的紧密联系以及随后将它们分解为不可约部分及其代数分类理论，为这些对象提供了坚实的基础。在第一部分中，我们回顾了将自缩放锥体家族描述为一组对称锥体的特征，并发展了关于自缩放障碍的原始-对偶对称观点，这是第二作者首先发现的结果。然后，我们用一个简短的证明证明，任何尖的凸锥体都会以独特的方式分解为不可约分量的直接和，这个结果也可能具有独立的意义。然后，我们继续证明任何自缩放势垒函数都会以一种本质上独特的方式分解为在不可约分量上定义的自缩放势垒的直接和
. Self-scaled barrier functions on self-scaled cones were axiomatically introduced by Nesterov and Todd in 1994 as a tool for the construction of primal–dual long-step interior point algorithms. This paper provides firm foundations for these objects by exhibiting their symmetry properties, their close ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and their algebraic classification theory. In the first part we recall the characterization of the family of self-scaled cones as the set of symmetric cones and develop a primal–dual symmetric viewpoint on self-scaled barriers, results that were first discovered by the second author. We then show in a short, simple proof that any pointed, convex cone decomposes into a direct sum of irreducible components in a unique way, a result which can also be of independent interest. We then proceed to showing that any self-scaled barrier function decomposes, in an essentially unique way, into a direct sum of self-scaled barriers defined on the irreducible components