Parallel Transport Cone Programming: A Manifold-Geometric Framework for Constrained Optimization

Traditional cone programming has wide applications in convex optimization, signal processing, quantum information, and machine learning. Its standard form is min c^T x, s.t. A x = b, x ∈ K. However, this model assumes the variable space is Euclidean space, which cannot characterize the geometric structure of variables located on non-Euclidean manifolds. Parallel transport in differential geometry provides a tool for aligning vectors in different tangent spaces on a manifold. This paper proposes a novel optimization theory—Parallel Transport Cone Programming (PTCP)—which combines cone programming with the manifold parallel transport mechanism to achieve constrained optimization modeling in the geometric space of a manifold. This paper systematically establishes the PTCP mathematical model, optimality conditions, and algorithmic solution framework, analyzes convergence, complexity, and geometric consistency, and explores its potential applications in geometric deep learning, quantum information optimization, information geometry, and high-dimensional statistical learning, providing a unified theoretical foundation for manifold constrained optimization.