On the calculus of limiting subhessians

We discuss various qualification assumptions that allow calculus rules for limiting subhessians to be derived. Such qualification assumptions are based on a singular limiting subjet derived from a sequence of efficient subsets of symmetric matrices. We introduce a new efficiency notion that results in a weaker qualification assumption than that introduced in Ioffe and Penot (Trans Amer Math Soc 249: 789-807, 1997) and prove some calculus rules that are valid under this weaker qualification assumption.