On the Calculus of Limiting Subhessians

Abstract

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.

On partial smoothness, tilt stability and the $${\mathcal {VU}}$$-decomposition

“…Denote the recession directions of a convex set C by 0 + C. Noting that Q, uv T = v T Qu one may see the motivation for the introduction of the rank-1 support in (9). The rank-1 support is given by q (A) (u, v) := sup Q, uv T | Q ∈ A for a subset A ⊆ S (n).…”

“…Many text book examples of these quantities can be easily constructed. Moreover there exists a robust calculus for the limiting subjet [9,23]. Furthermore as noted in example 51 of [11] the qualification condition for the sum rule for the limiting subjet can hold while for the same problem the basic qualification condition for the sum rule for the limiting (first order) subdifferential can fail to hold.…”

“…This is no small part to the fact that it consists only of symmetric matrices. This discussion has been carried out in quite a bit of detail in [9]. From convex analysis we know that the barrier cone (the points at which the support function is finite valued) is polar to the recession directions.…”

“…From convex analysis we know that the barrier cone (the points at which the support function is finite valued) is polar to the recession directions. In [9,Lemma 14] it is shown that for a rank-1 representer (using the Frobenious inner product on S (n)) this corresponds to (0 Lemma 24] it is shown that P(b 1 (A)) • ∩ P(n) = P(b 1 (A) ⊥ ). Denoting U 2 := b 1 (A) and V 2 = (U 2 ) ⊥ we deduce that P(V 2 ) = (0 + A) ∩ P(n).…”

“…The limiting pairs (z,Q) are contained in the so called "subjet" [5] and the second order componentsQ associated with a given z ∈ ∂f (x) are contained in the limiting subhessian ∂ 2 f (x,z), [34,23,10]. These have been extensively studied and possess a robust calculus similar to that which exists for the limiting subdifferential [23,9]. One can view the "best curvature" approximation in the direction h for the function f at (x,z) to be q(∂ 2 f (x,z))(h) := sup{ Qh, h | Q ∈ ∂ 2 f (x,z)}, where we denote by u, h the usual Euclidean inner product of two vectors u, h ∈ R n .…”