Unified Conic Programming Optimization under Information-Geometric Fisher Metric

This paper proposes a unified optimization theory that deeply integrates traditional conic programming (CP) with Fisher's metric (FM) in information geometry, constructing a conic programming optimization model on statistical manifolds. By introducing Fisher's information manifold metric, Lagrange KKT conditions, and conic projection mapping, a closed unified optimization equation is proposed, achieving adaptive search and global convergence guarantee in high-dimensional convex constraint spaces. Theoretically, this method can be extended to high-dimensional constraint optimization problems of semi-definite conic problems, covariance optimization, and deep probabilistic models, providing an original breakthrough for optimization theory. This model solves the problems of traditional conic programming easily getting trapped in local optima, non-smooth boundaries, and distorted gradient directions under complex high-dimensional constraints.