AI 中文总结
研究希尔伯特空间中可行集为仿射横截希尔伯特子流形的优化问题，通过开发工具包、推导导数表达式、引入投影诱导黎曼度量等建立理论基础，提出算法设计理论框架并展示相关算法。
AI 中文摘要
本文为希尔伯特空间中的一般优化问题建立理论基础，其可行集是由非线性流形与仿射子空间相交给出的仿射横截希尔伯特子流形。我们开发了几何和分析工具包，以确保迭代的可行性。还推导了提升目标泛函导数的弱形式表达式。引入投影诱导的黎曼度量，其诱导范数与环境范数一致等价，使投影诱导梯度成为具有显式公式的精确黎曼梯度。这为算法设计提供了实用的变度量框架，同时保留收敛分析所需的几何结构。有了这些理论基础，就可以应用欧几里得空间中的标准技术来设计算法。我们还通过展示具有严格收敛分析的黎曼线搜索和信赖域算法，提出了仿射横截希尔伯特子流形上算法设计的理论框架。
英文摘要
In this paper, we establish the theoretical foundations for the generic optimization problem in a Hilbert space whose feasible set is an affine-transversal Hilbert submanifold given by the intersection of a nonlinear manifold and an affine subspace. We develop the geometric and analytical toolkit, such as the tangent-space characterization, projection operators, and implicit retraction operators, to essentially ensure the feasibility of iterates for algorithmic design. We also derive weak-form expressions for the derivatives of lifted objective functionals. In particular, we introduce a projection-induced Riemannian metric whose induced norm is uniformly equivalent to the ambient norm, under which the projection-induced gradient becomes an exact Riemannian gradient with an explicit formula. This construction replaces the implicit tangent-space Riesz representation underlying classical Riemannian optimization with directly computable operator evaluations, yielding a practical variable-metric framework for algorithmic design while preserving the geometric structure required for convergence analysis. With these theoretical foundations, it becomes possible to apply standard techniques in Euclidean spaces to design algorithms with strictly feasible iterates for the optimization problem on an affine-transversal Hilbert submanifold. We also propose the theoretical frameworks for algorithmic design on affine-transversal Hilbert submanifolds by showcasing the Riemannian line-search and trust-region algorithms with rigorous convergence analysis.
Comments46 pages
