In second-order algorithms, we investigate the relevance of the constant rank of the full set of active constraints in ensuring global convergence to a second-order stationary point. We show that second-order stationarity is not expected in the non-constant rank case if the growth of the so-called tangent multipliers, associated with a second-order complementarity measure, is not controlled. We then investigate how these parameters should be controlled in order for the second-order information to remain present. Since no algorithm directly controls the growth of tangent multipliers, we argue that there is a theoretical limitation of present algorithms in finding second-order stationary points beyond the constant rank case.
View Some theoretical limitations of second-order algorithms for smooth constrained optimization