KL-Projected Conic Optimization: A Unified Framework for Constrained Divergence Minimization

This paper proposes a novel optimization framework that organically integrates traditional general conic programming (CP) with Kullback-Leibler projection (KLP) from information theory, constructing a KLP optimization model under conic constraints. This model, while preserving the convexity of the conic structure, introduces relative entropy constraints to achieve optimal approximation of probability distributions or non-negative vectors within the conic domain. By systematically establishing theoretically self-consistent optimization equations, this paper proves that the proposed method possesses global convergence, uniqueness, and structure fidelity under general convex cones. Furthermore, it presents generalization forms and iterative solution algorithms for different cone types, providing a unified mathematical tool for high-dimensional convex optimization, sparse probabilistic inference, and distributed optimization. This model overcomes the limitation of traditional conic programming with only linear objectives, while extending the feasible region of KLP projection to complex conic spaces, possessing convexity, information fidelity, and interpretability.