Certified Feasibility for Inexact Riemannian Frank–Wolfe on Strongly Convex Sets

We study constrained optimization on Riemannian manifolds where every iterate must remain feasible despite numerical approximation of the updates. Riemannian Frank–Wolfe preserves feasibility by moving along geodesics within a geodesically convex constraint set. Numerical approximation and finite-precision storage, however, can destroy this guarantee even when the output remains a valid manifold point. For numerical maps, we derive an acceptance rule combining a geometric tolerance from strong convexity of the feasible set with a quadratic descent tolerance; given feasible oracle targets, valid error bounds, and the stated oracle-accuracy and step-schedule conditions, accepted iterates remain feasible and retain the standard sublinear bound for smooth geodesically convex objectives. We then derive an intrinsic tangent-cone margin and, under smooth logarithms, an exact-oracle scaling inequality with a multiplicative conditioning factor. We use this inequality to transfer known faster rates when the gradient norm is bounded away from zero. We derive sharp rotation-ball formulas that make the geometric tolerances explicit. Finally, for a fixed-sub-step Euler integrator in a rotation chart, we prove that mantissa precision logarithmic in the inverse target accuracy suffices, given an exact feasible oracle, the stated primitive-error model, and sufficient available precision. Rotation experiments show that the acceptance rule maintains feasibility and recovers reference accuracy under low-precision storage through accurate fallback.