论文标题
在部分有限凸编程中的核心和准相对内部的注释
A Note on Cores and Quasi Relative Interiors in Partially Finite Convex Programming
论文作者
论文摘要
最小化熵功能受到线性约束的问题是部分有限的凸编程的一个有用示例。在1990年代，Borwein和Lewis提供了广泛而易于验证的条件，以确保针对此类问题的二元性。他们的方法是在相关的无限维集合的准搭配内部构建功能，该函数确保存在相关有限维集的核心中的一个点。我们重新审视了这个问题，并通过直接吸引核心的定义来提供替代证明，而不是依靠准融合性内部的任何属性。我们的方法承认了Borwein和Lewis框架中线性独立性要求的轻微放松，这使我们能够使用某些分段定义的力矩功能，因此无法通过其条件。我们提供了这样一个计算的示例，该示例说明了如何使用这种放松来驯服吉布斯现象时，当基础数据是不连续的。放松说明了我们通过从有限维和无限维方面解决部分有限问题来获得的理解。这两种方法的比较是有益的，因为这两种证明都是建设性的。
The problem of minimizing an entropy functional subject to linear constraints is a useful example of partially finite convex programming. In the 1990s, Borwein and Lewis provided broad and easy-to-verify conditions that guarantee strong duality for such problems. Their approach is to construct a function in the quasi-relative interior of the relevant infinite-dimensional set, which assures the existence of a point in the core of the relevant finite-dimensional set. We revisit this problem, and provide an alternative proof by directly appealing to the definition of the core, rather than by relying on any properties of the quasi-relative interior. Our approach admits a minor relaxation of the linear independence requirements in Borwein and Lewis' framework, which allows us to work with certain piecewise-defined moment functions precluded by their conditions. We provide such a computed example that illustrates how this relaxation may be used to tame observed Gibbs phenomenon when the underlying data is discontinuous. The relaxation illustrates the understanding we may gain by tackling partially-finite problems from both the finite-dimensional and infinite-dimensional sides. The comparison of these two approaches is informative, as both proofs are constructive.
