不可约对称锥的自缩放障碍函数
Self-scaled barriers for irreducible symmetric cones
AI总结：
本文分类了不可约对称锥上的所有自缩放障碍函数，证明它们只是通用障碍函数的位似变换，并结合分解定理完成了代数分类理论。
摘要：
自缩放障碍函数是对称锥上线性优化内点法理论中的基本对象，其中线性规划和半定规划是特例。我们对不可约对称锥上的所有自缩放障碍函数进行分类，并证明这些函数只是通用障碍函数的位似变换。结合自缩放障碍函数的分解定理，这完成了这些函数的代数分类理论。在向读者介绍与该问题相关的概念并追溯该主题的历史之后，我们首先在半定规划这一重要特例中从基本原理推导出我们的结果。然后，通过引用欧几里得若尔当代数理论的结果，将这些论证推广到不可约对称锥。
英文摘要：
Self-scaled barrier functions are fundamental objects in the theory of interior-point methods for linear optimization over symmetric cones, of which linear and semidefinite programming are special cases. We are classifying all self-scaled barriers over irreducible symmetric cones and show that these functions are merely homothetic transformations of the universal barrier function. Together with a decomposition theorem for self-scaled barriers this concludes the algebraic classification theory of these functions. After introducing the reader to the concepts relevant to the problem and tracing the history of the subject, we start by deriving our result from first principles in the important special case of semidefinite programming. We then generalise these arguments to irreducible symmetric cones by invoking results from the theory of Euclidean Jordan algebras.
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