Abstract Several logarithmic-barrier interiors-point methods are now available for solving two-stage stochastic optimization problems with recourse in the finite-dimensional setting. However, despite the genuine need for studying such methods in general spaces, there are no infinite-dimensional analogs of these methods. Inspired by this evident gap in the literature, in this paper, we propose logarithmic-barrier decomposition-based interior-point algorithms for two-stage stochastic linear optimization problems with recourse in a Hilbert space. We study the fundamental properties of the logarithmic barrier associated with the recourse function of our problem setting. The novelty of our algorithms is that their iteration complexity results are independent on the choice of the underlying Hilbert space. In other words, after applying the obtained fundamental properties to our problem setting, the iteration complexity results obtained for the short- and long-step algorithms coincide with the best-known estimates in the finite-dimensional case.

中文翻译：

---

希尔伯特空间中随机线性优化的对数势垒分解内点方法

摘要 现在有几种对数势垒内部点方法可用于求解有限维设置中具有追索权的两阶段随机优化问题。然而，尽管确实需要在一般空间中研究这些方法，但这些方法没有无限维的类似物。受文献中这一明显差距的启发，在本文中，我们提出了基于对数障碍分解的内点算法，用于在希尔伯特空间中具有资源的两阶段随机线性优化问题。我们研究与我们的问题设置的追索函数相关的对数障碍的基本属性。我们算法的新颖之处在于它们的迭代复杂度结果与底层希尔伯特空间的选择无关。换句话说，