Intrinsic Riemannian Conic Optimization: A Unified Framework for Conic Constraints and Second-Order Geometry on Manifolds

Cone programming is a core component of modern convex optimization theory, uniformly describing important problems such as linear programming, second-order cone programming, and semidefinite programming. However, classical cone programming assumes a Euclidean variable space, thus neglecting the non-Euclidean geometric structures of variables in many practical problems. This paper proposes a new optimization theoretical framework—Intrinsic Riemannian Conic Optimization (IRCO). Its core idea is to extend the cone constraint structure to the tangent space of Riemannian manifolds and utilize the Riemannian Hessian to describe the second-order optimality structure. Within this framework, we: 1. Introduce the concept of a tangent cone field 2. Establish the Riemannian cone Lagrangian function 3. Derive the new Riemannian cone KKT equations 4. Construct the Riemannian cone duality theory 5. Design the Riemannian primordial-dual Newton algorithm Theoretical analysis shows that, under the geodesic Slater condition, the IRCO problem satisfies strong duality, and the proposed algorithm exhibits local quadratic convergence. This framework provides a new geometric perspective for optimization theory and establishes a systematic connection between cone programming and Riemannian geometry.