Cone–Manifold Programming: A Unified Optimization Framework for Conic Geometry and Stiefel Manifold Constraints

Conic programming and manifold optimization constitute two important research directions in modern optimization theory. Conic programming is based on convex geometry, while manifold optimization relies on Riemannian geometry. Although both have wide applications in machine learning, signal processing, and statistical learning, there is currently a lack of an optimization theoretical framework that unifies these two types of structures. This paper proposes a new optimization model: Cone–Manifold Programming (CMP) This model unifies convex cone constraints and Stiefel manifold orthogonality constraints into a new geometric structure— Cone–Manifold Geometry. Specifically, we study the following optimization problem: min c^T vec(X) s.t. A vec(X) = b X ∈ K X^T X = I where K is a closed convex cone. The main contributions of this paper include: 1. Proposing the Cone–Manifold feasible region structure 2. Constructing the Cone–Manifold projection operator 3. Deriving new Cone–Manifold KKT conditions 4. Establishing the Cone–Manifold duality theory 5. Proposing the Cone–Manifold primal dual algorithm 6. Proving the global convergence and complexity bounds of the algorithm This theory provides a new unified perspective for optimization problems with structural constraints..