AI 中文总结
本文针对无严格可行性的半定可行性问题，构建了推导径向型Hölder误差界的系统框架，补充了Sturm的经典结果，还将其应用于半定规划最优性系统并验证了渐近紧性。
AI 中文摘要
本文针对不假设严格可行性（Slater条件）的半定可行性问题，构建了一套推导显式误差界的系统框架，而现有结果在该场景下十分有限。我们的主要技术贡献是引入了径向型Hölder误差界，其中误差界常数通过径向模函数显式依赖于参考矩阵的范数。结合面约化方法与最新提出的面残差函数，我们得到了这些模函数的显式表达式，进而在不施加任何约束规范的情况下，得到了定性的径向型Hölder误差界。我们的结果补充了Sturm的经典工作，为Sturm在给定大小的有界集合上的局部Hölder误差界中涉及的常数提供了显式估计。我们进一步分析了这些误差界在基础矩阵空间维度增长时的渐近行为，确定了它们在何种情况下可达到渐近紧性（误差仅相差一个与维度无关的常数）。作为应用，我们通过将半定规划的最优性系统重构为可行性问题，为其建立了显式误差界，而该场景下Slater条件通常不成立。在普遍满足的严格互补条件下，我们推导了不假设常规解唯一性要求的径向型误差界，并通过一个显式例子证明了其渐近紧性。
英文摘要
In this paper, we develop a systematic framework for deriving explicit error bounds for semidefinite feasibility problems without assuming strict feasibility (Slater's condition), a setting in which existing results are limited. Our main technical contribution is the introduction of radial-type Hölder error bounds, where the error bound constant depends explicitly on the norm of the reference matrix through radial modulus functions. By combining facial reduction with recently developed facial residual functions, we obtain explicit descriptions of these modulus functions, yielding qualitative radial-type Hölder error bounds \emph{without imposing any constraint qualifications}. Our results complement the classical work of Sturm by providing explicit estimates for the constants involved in Sturm's local Hölder error bounds over bounded sets with a given size. We further analyze the asymptotic behavior of these bounds as the dimension of the underlying matrix space grows, identifying regimes in which they can be asymptotically tight up to a dimension-free constant. As an application, we establish explicit error bounds for the optimality system of semidefinite programs by reformulating them as feasibility problems, a setting where Slater's condition typically fails. Under the generically satisfied strict complementarity condition, we derive radial-type error bounds without assuming the usual solution uniqueness requirement, and demonstrate their asymptotic tightness through an explicit example.
