发表机构
Xidian University（西安电子科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究优化几何动力学中受限动态几何复杂度，通过发展相关理论，如单调性和子流形距离原理等，将结构化预处理器问题转化为几何问题，还给出低秩谱模型等诊断接口及相关工作流程。
AI 中文摘要
优化几何动力学将优化器状态视为演化的几何结构。其全正定二次基准给出了在允许任意度量时降低条件数所需的最小仿射不变变形。本文用当前符号记录该基准，并发展了受限动态几何复杂度：当度量限制在指定族时达到目标条件数类的内在证书距离。主要证明结果包括单调性和子流形距离原理、作为线性矩阵不等式可行性问题的对角和块可达性、精确的二维对角复杂度公式、带正规方程的仿射不变克罗内克投影定理、可计算的失配证书、阿米尔乔求解器收敛、辅助自条件K目标界以及通过精确的克罗内克 - 洛纳三明治可达性条件得到的黑塞相对候选证书等。还介绍了低秩谱模型等作为诊断接口。最终语言将结构化预处理器问题转化为几何距离、可达性和证书问题。存储库包含确定性玩具和合成工作流程，可检查小二次实例上的对角表达式差距等。
英文摘要
Structured preconditioners restrict optimization to a small family of positive metrics, but endpoint condition-number reachability does not measure the geometric effort required to reach a useful metric. We formulate this effort as a path-space value problem. Restricted dynamic geometric complexity is the least affine-invariant length of an admissible metric path whose endpoint reaches a Hessian-relative generalized-eigenvalue condition target. Path elimination gives an exact min-plus semigroup and Bellman principle, while fixed-horizon kinetic energy is exactly squared complexity divided by twice the horizon. The main response result is global on a Hadamard state space: geodesic convexity produces a smooth intervention-cube path branch and a uniformly coercive Jacobi form, while one Green inverse generates the value Hessian, two-sided force-to-curvature bounds, exact Möbius effects, and arbitrary prescribed finite-order responses. For the hard condition target, a bordered Jacobi--KKT theorem differentiates the moving projection endpoint and multiplier on every regular active spectral stratum; its indefinite inverse also explains why hard-target interactions need not share the unconstrained sign. The theory specializes to affine-invariant positive-definite geometry. A determinant-one two-dimensional diagonal model has an exact target interval, a closed-form forced path, and a strictly negative-definite interaction matrix. A moving diagonal Hessian gives a closed-form hard-target projection, multiplier, and pair effects of either sign, while a coordinate-sequential three-dimensional protocol yields an exact path metric strictly larger than the ambient projection distance. Thus the global Green and bordered hard-target responses are explicit laws of restricted metric-path elimination built on Bellman composition.
Comments32 pages, 1 table
