{"ID":2877966,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.18764","arxiv_id":"2508.18764","title":"The Geometry of Constrained Optimization: Constrained Gradient Flows via Reparameterization: A-Stable Implicit Schemes, KKT from Stationarity, and Geometry-Respecting Algorithms","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2508.18764","url_pdf":"https://arxiv.org/pdf/2508.18764v2","authors":"[\"Valentin Leplat\"]","published":"2025-08-26T07:45:47Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}