Calculus for Directional Limiting Normal Cones and Subdifferentials

Abstract

The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.

Directional Quasi-/Pseudo-Normality as Sufficient Conditions for Metric Subregularity

“…Recently based on the concept of the directional limiting normal cone, the following directional version of the limiting subdifferential was introduced in [2].…”

“…Recall that in optimization we call a multiplier abnormal if it is a multiplier corresponding to an optimality system where the objective function vanishes. Assuming P is continuously differentiable (C 1 ), if the no nonzero abnormal multiplier constraint qualification (NNAMCQ) holds, i.e., there is no nonzero abnormal multiplier ζ such that 0 = ∇P (x) * ζ, ζ ∈ N Λ (P (x)), (2) where N Λ (·) is the limiting normal cone, ∇P denotes the Fréchet derivative of P , and * denotes the adjoint, then the Mordukhovich's criteria for metric regularity (see, e.g., [46,Theorem 9.40]) holds and so does metric subregularity. These two criteria are relatively strong since they are actually sufficient conditions for stronger stability concepts.…”

“…Another direction in the effort of weakening the NNAMCQ is to add some extra conditions to (2). In the case where P is continuously differentiable atx, we say that quasinormality and pseudo-normality hold atx if there exists no ζ = 0 such that (2) holds and…”

“…with a i ∈ R n and ρ in the range (0, ∞]; see, e.g., [38,40,41]. Apparently when ρ takes a value in the interval (1,2), the optimality condition of minimizing function (3) with respect to x can be described by 0 ∈ ∇ x f (x, b)+∂g(x), where ∂g(x) denotes a certain subdifferential of g at x, which can be reformulated as P (x) ∈ Λ where P (x) is continuous. Therefore, thanks to the equivalence between calmness of different reformulations established in [15,Proposition 3], our results can be used to study the calmness of the optimality condition system of minimizing (3) without imposing an unnecessarily stronger condition.…”

“…directional subdifferentials; see[2]) Let f : X → [−∞, +∞] andx be a point where f is finite. Then the limiting subdifferential of f atx in direction (u, ζ) ∈ X ×R is defined as…”