Orthogonal Conic-Manifold Optimization: A Unified Framework and Optimality Equation for High-Dimensional Conic Constraints

This paper proposes an original optimization theory—Orthogonal Conic-Manifold Optimization (OCMO)—which combines **general cone programming (CP)** with the manifold Newton method (MN). By constructing the local tangent space and Hessian projection of the orthogonal conic manifold, it achieves a second-order exact iterative solution to high-dimensional cone-constrained optimization problems. This paper presents a novel mathematical model—the Orthogonal Conic-Manifold Optimality Equation (OCMOE)—and provides a detailed derivation of the iteration strategy, convergence conditions, and theoretical properties, forming a unified theoretical framework directly applicable to high-dimensional convex and non-convex cone-constrained problems. This theory serves as a new mathematical tool for cone-constrained optimization, possessing high theoretical originality and application potential.