Primal-Dual Manifold Newton Method for Semidefinite Programming: Theory and Applications

This paper proposes a novel optimization method that deeply integrates Dual Semidefinite Programming (DDP) with the Manifold Newton Method, constructing a Primal-Dual Newton Flow on Manifolds for dual semidefinite optimization. Compared with traditional SDP and manifold optimization methods, this method achieves second-order convergence while ensuring global convergence. Furthermore, by introducing adaptive manifold mapping and dual gap adjustment mechanisms, it effectively enhances the ability to handle uncertain constraints and non-smooth objective functions. This paper demonstrates the advantages of this method from three levels: theoretical, algorithm design, and numerical experiments, providing a new theoretical tool for high-dimensional nonlinear optimization problems.