Escaping High-order Saddles in Policy Optimization for Linear Quadratic Gaussian (LQG) Control

The paper’s Theorem 2 proves that for a non-minimal stationary point K̃ built by augmenting a minimal realization with a symmetric Hurwitz Λ, the Hessian of the full-order LQG cost is indefinite whenever eig(−Λ) avoids the zero set of a certain transfer G(s); it further deduces the almost-sure strict-saddle property for random Λ and that global optimality forces G ≡ 0. The candidate solution derives an explicit bilinear form for the Hessian along the same coupling directions and reaches the same three conclusions via the same block-augmentation/Lyapunov–Sylvester machinery. The two arguments align on definitions, structure, and logic; the model’s derivation is slightly more explicit (resolvent/Sylvester and a sum over eigenvalues) but substantively the same.