标题： 约束优化的几何：通过参数化约束梯度流：A-稳定隐式格式，从平稳性得到KKT，以及尊重几何的算法 显示英文标题

标题： The Geometry of Constrained Optimization: Constrained Gradient Flows via Reparameterization: A-Stable Implicit Schemes, KKT from Stationarity, and Geometry-Respecting Algorithms

摘要： 梯度流（GF）观点统一并阐明了优化算法，但大多数GF分析集中在无约束设置上。 我们通过（i）用雅可比矩阵在边界（正则/盒子）上消失或在单纯形上具有秩$n-1$的映射重新参数化可行集（Fisher-Shahshahani 算子），（ii）推导出在参数空间中诱导可行原始动力学的流，（iii）使用A稳定的隐式方案进行离散化（向量域上的后向欧拉；Stiefel上的可行Cayley），由稳健的内部循环求解（修改的高斯-牛顿或KL近似/负熵牛顿-KKT求解器），以及（iv）证明动态的平稳性意味着KKT，互补松弛性来源于一个简单的运动学机制（由雅可比矩阵消失或单纯形上的Fisher-Shahshahani算子引起的零法向速度）。 我们还处理了Stiefel流形，其中黎曼平稳性与KKT一致。 该理论为每种约束类提供了高效且尊重几何的算法，具有单调下降且没有步长限制。 我们简要讨论了A稳定性，并展示了数值测试（NNLS、单纯形和盒子约束的最小二乘问题以及Stiefel）表明隐式方案的稳定性、准确性和运行效率。 显示英文摘要

摘要： Gradient-flow (GF) viewpoints unify and illuminate optimization algorithms, yet most GF analyses focus on unconstrained settings. We develop a geometry-respecting framework for constrained problems by (i) reparameterizing feasible sets with maps whose Jacobians vanish on the boundary (orthant/box) or have rank $n-1$ on the simplex (the Fisher--Shahshahani operator), (ii) deriving flows in parameter space that induce feasible primal dynamics, (iii) discretizing with A-stable implicit schemes (backward Euler on vector domains; feasible Cayley on Stiefel) solved by robust inner loops (modified Gauss--Newton or a KL-prox/negative-entropy Newton--KKT solver), and (iv) proving that stationarity of the dynamics implies KKT, with complementary slackness arising from a simple kinematic mechanism (zero normal speed induced by a vanishing Jacobian or by the Fisher--Shahshahani operator on the simplex). We also treat the Stiefel manifold, where Riemannian stationarity coincides with KKT. The theory yields efficient, geometry-respecting algorithms for each constraint class, with monotone descent and no step-size cap. We include a brief A-stability discussion and present numerical tests (NNLS, simplex- and box-constrained least squares, and Stiefel) demonstrating stability, accuracy, and runtime efficiency of the implicit schemes.