Calculus of directional subdifferentials and coderivatives in Banach spaces

Calculus for Directional Limiting Normal Cones and Subdifferentials

“…Our approach to directional limiting subdiferentials differs from the one established in [3,11,16], where it is either defined or equivalently described as a limit of regular subdiferentials. In the finite dimensional setting these definitions read as follows.…”

“…Note that inner calmness of S at (ȳ, x) ∈ gph S w.r.t. dom S in direction v exactly corresponds to the directional inner semicompactness of S at (ȳ, x) ∈ gph S in direction v from [16,Definition 4.4]. In literature one can find also several other names for this property, such as, e.g., Lipschitz lower semicontinuity [14] or recession with linear rate [12].…”

“…The directional versions of inner semicompactness and semicontinuity are obtained by restricting our attention to y k converging to ȳ from some direction v. We point out here that in [16,Definition 4.4] the authors defined the directional versions of inner semicompactness and semicontinuity in such a way that it allows them to find a suitable direction h, i.e., they control the rate of convergence x k → x by requiring the difference quotients (x k − x)/t k either to be bounded or to converge to some prescribed h. We believe, however, that it is not very suitable to call such properties semicompactness and semicontinuity, as those requirements are clearly much stronger and they are not implied by their non-directional counterparts, as also the authors admit.…”

“…In particular, in [4] one finds, apart from some elementary rules, formulas for directional limiting normal cones to unions of convex polyhedral sets and in [8], [9] second-order chain rules have been derived which enable us to compute directional limiting coderivatives to normal-cone mappings associated with various types of sets. Further, [16] contains several rules of the directional calculus even in Banach spaces, in [19] the authors proved a special coderivative sum rule and in [20] the situation has been examined when one has to do with the Carthesian products of sets and mappings.…”

“…It was pointed out in[16] that in[3] the condition f (x + t k u k ) → f (x) was omitted from the definition and the same holds true for the definition from[11]. Such definition leads e.g.…”