active set method guaranteed convergence

I'm using Active Set Method to solve a nonlinear optimization function minimizing a convex function over a polyhedron of 2 linear inequalities starting at an interior point $$x_o$$ At this point is it correct that the direction of travel can be anywhere insofar as it's a descent direction, ie the $$Z_+$$ basis of the null space is $$I$$ because there are 0 active constraints?

Whenever I perform the algorithm by hand, the Reduced Newton Direction with maximum step $$\alpha$$ brings me to a feasible solution that meets local optimum conditions and is on the boundary of constraint C1, but it is not the global solution as the better minimum actually exists on the boundary of the 2nd constraint C2 which I am unable to reach once I'm on the boundary of C1.

Thus my question is, based on where you start the algorithm $$x_0$$ is global convergence guaranteed using active set method with Reduced Newton Direction or does it heavily depend on the starting point to find a local min that satisfies the algorithm's stopping conditions?