Riemannian Convexity in Programming (II)

The book [7] emphasizes three relevant aspects: first, the fact that the notion of convexity is strongly metric-dependent either through geodesics or through the Riemannian connection; second that Riemannian convexity of functions is a coordinate-free concept, and consequently it can be easily connected with symbolic computation; third, that the Riemannian structure is involved essentially in formulating and solving programs by means of induced distance, geodesics, Riemannian connection, sectional curvature, etc. The preceding arguments justify the effort to generalize the optimization theory on Euclidean spaces to the Riemannian manifolds. The generalization is obtained by selecting a suitable Riemannian metric, by passing from vector addition to the exponential map, by changing the search along straight lines with a search along geodesics, and by using covariant differentiation instead of partial differentiation. §1 shows that some difficulties appearing in the free-minimization problems belong to a wrong understanding of the suitable Riemannian structure of the space. §2 deals with Newton algorithm on Riemannian manifolds for finding zeros of a C∞ vector field or generally for a C∞ tensor field. §3 describes the path of centers attached to a convex program on a Riemannian manifold and analysis the monotonicity of the objective function along this curve. §4 gives theorems regarding the Newton method near the path of centers of a convex program (one unit Newton step stays inside the feasible set, quadratic convergence results, upper bound for the difference of two Huard distance function values, etc), using simultaneously the original Riemannian metric and a Hessian Riemannian metric. §5 gives upper bounds for the total number of outer iterations and inner iterations needed by the center algorithm on a Riemannian manifold. The theorems in §3-5 have their origin in the Euclidean variants exposed in [1], [2] and in the Riemannian point of view about convex programming developed in [7]. 1991 Mathematics Subject Classification: 49M45,49M15