Cone-Induced Riemannian Optimization: A Unified Geometric Framework for Conic Programming

Cone programming is a core framework of modern convex optimization, its standard form being: min c^T x s.t. Ax = b x ∈ K where K is a closed convex cone. On the other hand, Riemannian optimization studies gradient methods on smooth manifolds, where the gradient is defined as: grad f(x) = G(x)^(-1) ∇f(x) where G(x) is the Riemannian metric tensor. This paper proposes a new theoretical framework: Conic-Induced Riemannian Optimization (CIRO) The core idea is: to consider the interior of the cone, int(K), as a Riemannian manifold induced by the cone potential function, and to define optimization dynamics on this manifold. Within this framework, we propose a new unified optimization equation: dx/dt = -Π_K(x) ( H_K(x)^(-1) ∇f(x) ) where H_K(x) = ∇^2 Φ_K(x) is the Hessian of the cone potential function. This paper proves: 1. Feasibility Preservation Theorem 2. Monotonically Descent Theorem 3. Global Convergence Theorem 4. Complexity Boundary Theorem This framework unifies: Interior-point methods Riemannian gradient descent Conic optimization and provides a new geometric optimization perspective.