{"ID":2841221,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.12064","arxiv_id":"2511.12064","title":"Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopes","abstract":"In this paper, we address the optimization problem of minimizing $Q(df_x)$ over a Hadamard manifold ${\\cal M}$, where $f$ is a convex function on ${\\cal M}$, $df_x$ is the differential of $f$ at $x \\in {\\cal M}$, and $Q$ is a function on the cotangent bundle of ${\\cal M}$. This problem generalizes the problem of minimizing the gradient norm $\\|\\nabla f(x)\\|$ over ${\\cal M}$, studied by Hirai and Sakabe FOCS2024. We formulate a natural class of $Q$ in terms of convexity and invariance under parallel transports, and introduce a generalization of the gradient flow of $f$ that is expected to minimize $Q(df_x)$. For basic classes of manifolds, including the product of the manifolds of positive definite matrices, we prove that this gradient flow attains $\\inf_{x\\in {\\cal M}} Q(df_x)$ in the limit, and yields a duality relation. This result is applied to the Kempf-Ness optimization for GL-actions on tensors, which is Euclidean convex optimization on the class of moment polytopes, known as the entanglement polytopes. This type of convex optimization arises from tensor-related subjects in theoretical computer science, such as quantum functional, $G$-stable rank, and noncommutative rank.","short_abstract":"In this paper, we address the optimization problem of minimizing $Q(df_x)$ over a Hadamard manifold ${\\cal M}$, where $f$ is a convex function on ${\\cal M}$, $df_x$ is the differential of $f$ at $x \\in {\\cal M}$, and $Q$ is a function on the cotangent bundle of ${\\cal M}$. This problem generalizes the problem of minimi...","url_abs":"https://arxiv.org/abs/2511.12064","url_pdf":"https://arxiv.org/pdf/2511.12064v2","authors":"[\"Hiroshi Hirai\"]","published":"2025-11-15T07:09:13Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.DG\"]","methods":"[]","has_code":false}