The paper studies the asymptotic rate of convergence of the norm-relaxed method of feasible directions in the case of solving the problem of minimizing a strictly convex function subject to convex inequality constraints, all the problems' functions being of class \(C^2\). The main theorem shows that, either the norm-relaxed method of the feasible direction algorithm stops after a finite number of steps at the solution, or the infinite sequence of feasible solutions converges at least linearly to the optimum. In the case of an interior point optimum it is shown that the norm-relaxed FDM algorithm has a potential for superlinear convergence.