AI 中文总结
研究基于新正则性概念分析共辐射集，引入内逼近序列和径向轮廓函数两个工具，扩展分离理论，给出范数基存在新特征及标量化结果，通过建立误差界对分离残差变分解释，为平衡问题等提供条件和原理。
AI 中文摘要
我们基于一种新的正则性概念开发了一个用于分析共辐射集的新框架。该框架将经典理论扩展到了超出具有范数基的共辐射集的范围，并为分离和优化提供了统一的设置。我们引入了两个新工具：内逼近序列和径向轮廓函数。前者提供了一种正则性的序列方法，是正则共辐射集一般分离定理证明的关键要素，扩展了现有分离理论。后者为任意共辐射集定义，给出了范数基存在的新特征，以及在相关锥的实性假设下基于序列的正则性特征。它还产生了一个标量函数，可在无额外结构假设的情况下为近似效率带来标量化结果。最后，我们通过建立径向深度和度量误差界对分离残差进行变分解释。这些为平衡问题和变分不等式中的近似径向可行性提供了充分条件，以及残差零水平集上的精确惩罚原理。
英文摘要
We develop a new framework for the analysis of co-radiant sets based on a new notion of regularity. This framework extends the classical theory beyond co-radiant sets admitting a norm-base and provides a unified setting for separation and optimization. We introduce two new tools: inner approximating sequences and the radial profile function. The former provides a sequential approach to regularity and is the key ingredient in the proof of a general separation theorem for regular co-radiant sets, extending the existing separation theory. The latter is defined for arbitrary co-radiant sets and yields new characterizations of the existence of norm-bases, together with sequence-based characterizations of regularity under a solidness assumption on the associated cone. It also gives rise to a scalar function leading to scalarization results for approximate efficiency without additional structural assumptions on the underlying co-radiant set. Finally, we provide a variational interpretation of the separation residual by establishing radial-depth and metric error bounds. These yield sufficient conditions for approximate radial feasibility in equilibrium problems and variational inequalities, and an exact penalization principle on the zero level set of the residual.
