A note on the convergence of barrier algorithms to second-order necessary points

It has long been known that barrier algorithms for constrained optimization can produce a sequence of iterates converging to a critical point satisfying weak second-order necessary optimality conditions, when their inner iterations ensures that second-order necessary conditions hold at each barrier minimizer. We show that, despite this, strong second-order necessary conditions may fail to be attained at the limit, even if the barrier minimizers satisfy second-order sufficient optimality conditions.