SIAM Journal on Optimization, Volume 31, Issue 2, Page 1276-1298, January 2021.
Convex cones play an important role in nonlinear analysis and optimization theory. In particular, specific normal cones and tangent cones are known to be convex cones, and it is a crucial fact that they are useful geometric objects for describing optimality conditions. As important applications (especially, in the fields of optimal control with PDE constraints, risk theory, duality theory, vector optimization, and order theory) show, there are many examples of convex cones with an empty (topological as well as algebraic) interior. In such situations, generalized interiority notions can be useful. In this article, we present new representations and properties of the relative algebraic interior (also known as intrinsic core) of relatively solid, convex cones in real linear spaces (which are not necessarily endowed with a topology) of both finite and infinite dimension. For proving our main results, we are using new separation theorems where relatively solid, convex sets (cones) are involved. For the intrinsic core of the dual cone of a relatively solid, convex cone, we also state new representations that involve the lineality space of the given convex cone. To emphasize the importance of the derived results, some applications in vector optimization are given.

中文翻译：

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实线性空间中凸锥的本征核

SIAM优化杂志，第31卷，第2期，第1276-1298页，2021年1月。
凸锥在非线性分析和优化理论中起着重要作用。特别地，已知特定的法向圆锥和切圆锥为凸圆锥，而至关重要的事实是它们是用于描述最优性条件的有用几何对象。正如重要的应用程序（特别是在具有PDE约束的最优控制领域，风险理论，对偶理论，向量优化和阶数理论）所显示的那样，有许多内部为空（拓扑以及代数）的凸锥示例。在这种情况下，广义的内部概念可能会有用。在本文中，我们介绍了相对坚固的相对代数内部（也称为内在核）的新表示形式和性质，有限和无限维的实线性空间（不一定具有拓扑结构）中的凸锥。为了证明我们的主要结果，我们使用了新的分离定理，其中涉及相对较实的凸集（圆锥）。对于相对坚固的凸锥的双锥的本征核心，我们还陈述了涉及给定凸锥的线性空间的新表示形式。为了强调得出结果的重要性，给出了矢量优化中的一些应用。