Geodesic Cone Optimization: A Riemannian Approach to Conic Programming

Cone programming is a crucial component of modern optimization theory, uniformly characterizing linear programming, second-order cone programming, and semidefinite programming problems, providing a foundation for control theory, signal processing, and machine learning. However, traditional cone programming primarily relies on the Euclidean space assumption, neglecting the geometric structure of the feasible region itself, thus limiting a deeper understanding of high-dimensional, non-Euclidean constrained optimization problems. On the other hand, the geodesic equations in Riemannian geometry characterize the natural dynamic evolution and shortest path laws on manifolds, providing theoretical support for geometric optimization, information geometry, and natural gradient methods. This paper proposes a novel optimization framework—Geodesic Cone Optimization (GCO)—whose core idea is to embed cone programming into the structure of a cone manifold and utilize geodesic dynamics to construct optimization equations, naturally integrating the geometric structure of the feasible region, the gradient of the objective function, and the constraints. This paper first establishes a conical Riemannian metric model and derives the GCO dynamic equation: d^2 x^k / dt^2 + Gamma^k_ij(x) (dx^i/dt)(dx^j/dt) + alpha c_k + alpha beta (A^T (A x - b))_k = 0 Then it is proved that the model has local stability near the optimum and its asymptotic behavior strictly satisfies the KKT conditions of conical programming. Theoretical analysis shows that GCO not only unifies the theoretical foundations of the interior-point method, the natural gradient method, and the geometric Newton method, but also provides new geometric intuition and algorithm design ideas for high-dimensional constrained optimization problems. This paper also proposes a discretized algorithmic form of GCO and discusses its application potential in semidefinite programming, regularized machine learning, and deep learning geometric optimization. This research provides a new geometric perspective for optimization theory and lays the foundation for future high-dimensional non-Euclidean constrained optimization algorithms.