论文标题
欧几里得约旦代数的光谱集优化问题的换向原则
Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras
论文作者
论文摘要
拉米雷斯，塞格和索萨的换倒原则在欧几里得约旦代数的情况下得到了证明，当真实的值函数$ h $和光谱函数的总和$ h $ $ h $和光谱函数$φ$在频谱集$ e $中最大化/最大化，在$ h $ h $ h $ h $ h $ h $ h $ h $ h $ h $ h $ h $ h a $frécheta $ a是$ n is vertiv a $ $ $ $ $ $ $ $ $ $ $ a^a;在本文中，假设（$ h $）的衍生物存在亚级别，我们建立了“强操作员的通勤”关系：如果$ a $解决问题$ \ unterSet {e} {e} {\ max} \，（h+φ）如果$ e $和$ h $是凸，$ a $解决了问题$ \ underset {e} {\ min} \，h $，则$ a $ a $ a $ a $ a $ a $运算符在$ a $ a $ $ a $ a $ $ h $中的某些元素的负数。这些结果改善了线性$ h $的已知（运算符）通勤关系以及变异不平等问题的解决方案。我们通过几何换向原则建立这些结果，该原理不仅在欧几里得约旦代数中有效，而且在FTVN系统的更广泛的环境中也有效。
The commutation principle of Ramirez, Seeger, and Sossa proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function $h$ and a spectral function $Φ$ is minimized/maximized over a spectral set $E$, any local optimizer $a$ at which $h$ is Fréchet differentiable operator commutes with the derivative $h^{\prime}(a)$. In this paper, assuming the existence of a subgradient in place the derivative (of $h$), we establish `strong operator commutativity' relations: If $a$ solves the problem $\underset{E}{\max}\,(h+Φ)$, then $a$ strongly operator commutes with every element in the subdifferential of $h$ at $a$; If $E$ and $h$ are convex and $a$ solves the problem $\underset{E}{\min}\,h$, then $a$ strongly operator commutes with the negative of some element in the subdifferential of $h$ at $a$. These results improve known (operator) commutativity relations for linear $h$ and for solutions of variational inequality problems. We establish these results via a geometric commutation principle that is valid not only in Euclidean Jordan algebras, but also in the broader setting of FTvN-systems.
