A Unified Variational Theory of Cone-Induced Spectral Entropic Optimization

This paper establishes a novel optimization variational framework—the functional theory of cone-induced spectral entropy. On a general closed convex self-dual cone K, this theory constructs a spectral induced mapping T(x) through the second-order structure of the cone's internal barrier functions, and defines the spectral volume entropy functional log det T(x). We prove that this functional induces a natural self-dual variational structure within the cone and establish its existence and uniqueness theorem, strong duality theorem, limit degeneracy theorem, and Riemannian information geometric consistency theorem. This theory unifies cone programming and log det optimization, revealing that the essence of the interior-point method is the geometric flow of spectral entropy.