JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics Calculus for Directional Limiting Normal Cones and Subdifferentials

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.


Introduction
Since the early works of Mordukhovich, the limiting normal cone and the corresponding subdifferential belong to the central notions of variational analysis.They admit a rich calculus both in finite as well as in infinite dimensions and have been successfully utilized in a large variety of optimization and equilibrium problems, see [18], [17] and the references therein.In particular, these notions play an important role in stability and sensitivity issues, above all in analysis of various Lipschitzian stability notions related to multifunctions.Having been motivated by some of the above listed applications, in [11] the authors refined the original definitions by restricting the limiting process only to a subset of sequences used in the original definitions.This lead eventually to the notions of directional limiting normal cone and directional limiting subdifferential which have been further developed and utilized above all in the works authored or coauthored by Gfrerer [3,4,5,6,7,8,9,10].It turned out that these directional notions (together with the directional limiting coderivative) enable us indeed a substantially finer analysis of situations in which the estimates, provided by the standard calculus, are too rough and so the corresponding assertions are not very useful.This is, e.g., the case of verification of metric subregularity of feasibility mappings (calmness of perturbation mappings) related to constraint systems, which lead to new first-and second-order sufficient conditions for metric subregularity [3,4].They are now widely used as weak (non-restrictive) but yet verifiable qualification conditions [7,10,20].In [8] the authors used the directional limiting coderivative to establish new weak conditions ensuring the calmness and the Aubin property of rather general implicitly defined multifunctions and in [9] this technique has then been worked out for a class of parameterized variational systems.
Further, directional limiting coderivative appears in sharp first-order optimality conditions [3], entitled extended M-stationarity in [5], which provide a dual characterization of B-stationarity for disjunctive programs.
One can definitely imagine also numerous other problems of variational analysis in which the directional notions could be successfully employed.In all of them, however, one needs a set of rules enabling us to compute efficiently the directional normal cones and subdifferentials of concrete sets and functions in a similar way like in the standard generalized differential calculus of Mordukhovich.Some parts of such a calculus have already been conducted in connection with various results mentioned above.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.
The aim of this paper is to fill in this gap by building a systematic decent calculus of directional limiting normal cones and subdifferentials following essentially the lines of [17,Chapter 3] and [13].The structure is as follows.
In Section 2 we collect the needed definitions and present some auxiliary results used throughout the paper.Section 3 is devoted to the calculus of directional limiting normal cones.As the most important results we consider formulas for the directional limiting normal cone of the preimage and of the image of a set in a Lipschitz continuous mapping.These results have numerous important consequences.Section 4 concerns the calculus of directional limiting subdifferentials.Apart from the chain and sum rules we consider the case of value functions, distance functions, pointwise minima and maxima and examine also the partial directional limiting subdifferentials.Section 5 provides formulas for directional limiting coderivatives of compositions and sums of multifunctions together with some important special cases.In Section 6 we present some problems of variational analysis, where the usage of the directional limiting calculus leads to weaker (less restrictive) sufficient conditions or sharper (more precise) estimates.
Similarly as in [13], we have attempted to impose the "weakest" qualification conditions expressed mostly in terms of directional metric subregularity of associated feasibility mappings.Admittedly, these conditions are not always verifiable, but they can in many cases be replaced by stronger (more restrictive) conditions expressed in terms of problem data.
In the directional calculus one also meets a new specific difficulty associated with the computation of the images or the pre-images of the given direction in the considered mappings.This obstacle leads in some cases to more complicated rules or to additional qualification conditions.
It turns out, that the essential role in the calculus is played by Theorems 3.1 and 3.2, concerning the pre-image and the image of a set in a continuous/Lipschitz mapping.The basic ideas arising in these two statements appear in fact in almost all calculus rules throughout the whole paper.
The following notation is employed.The closed unit ball and the unit sphere in R n are denoted by B and S, respectively, while B r (x) := {x ∈ R n | x − x ≤ r}.The identity mapping is denoted by Id.Given a set Ω ⊂ R n , bd Ω stands for the boundary of Ω, i.e., the set of points whose every neighborhood contains a point of Ω as well as a point not belonging to Ω.Moreover, given also a point x, d Ω (x) denotes the distance from x to set Ω and P Ω (x) denotes the projection of x onto Ω.For a sequence x k , x k Ω → x stands for x k → x with x k ∈ Ω.Given a directionally differentiable function ϕ : R n → R m at x ∈ dom ϕ, ϕ (x; h) denotes the directional derivative of ϕ at x in direction h.Finally, following traditional patterns, we denote by o(t) for t ≥ 0 a term with the property that o(t)/t → 0 when t ↓ 0.

Preliminaries
We start by recalling several definitions and results from variational analysis.Let Ω ⊂ R n be an arbitrary closed set and x ∈ Ω.The contingent (also called Bouligand or tangent) cone to Ω at x, denoted by T Ω (x), is given by We denote by the Fréchet (regular) normal cone to Ω at x. Finally, the limiting (Mordukhovich) normal cone to Ω at x is defined by The Mordukhovich normal cone is generally nonconvex whereas the Fréchet normal cone is always convex.In the case of a convex set Ω, both the Fréchet normal cone and the Mordukhovich normal cone coincide with the standard normal cone from convex analysis and, moreover, the contingent cone is equal to the tangent cone in the sense of convex analysis. Note Consider an extended real-valued function f : R n → R and a point x the limiting (Mordukhovich) subdifferential of f at x, denoted by ∂f (x), is given by and the singular subdifferential of f at x is defined by

Denoting by epi
where the last relation holds if epi f is locally closed around (x, f (x)) or, equivalently, f is lower semicontinuous (lsc) around x, see e.g.[18,Theorem 8.9].
Given a multifunction M : R n ⇒ R m and a point (x, ȳ) i.e., D * M (x, ȳ)(η) is the collection of all ξ ∈ R n for which there are sequences (x k , y k ) → (x, ȳ) and when M is single-valued at x with M (x) = {ȳ}.Just as in case of subdiferentials and epigraphs, it is often important to have closed graphs of multifunctions.We say that M is outer semicontinuous (osc) at x if the existence of sequences x k → x and y k → y with y k ∈ S(x k ) implies y ∈ S(x) and we say that M is osc if it is osc at every point, which is equivalent to the closedness of gph M , see [18,Theorem 5.7].
For more details we refer to the monographs [17,18].Directional versions of these limiting constructions were introduced in [11] and [3] for general Banach spaces and later on equivalently reformulated for finite dimensional spaces in [5] in the following way.Given a direction u ∈ R n , the limiting normal cone to a subset Ω ⊂ R n at x ∈ Ω in direction u is defined by Note that by the definition we have The following simple lemma provides a hint about possible applications of directional limiting normal cones.
Proof.Inclusion ⊃ follows directly from definition.Now let ξ ∈ N Ω (x) and consider x k → x, ξ k → ξ with x k ∈ Ω and ξ k ∈ NΩ (x k ).If x k = x for infinitely many k we have ξ ∈ NΩ (x) due to closedness of NΩ (x).On the other hand if x k = x for infinitely many k, we set t k := x k − x and u k := (x k − x)/ x k − x and by passing to a subsequence we assume (t k ) ↓ 0 and u k → u ∈ S. Since x k = x + t k u k ∈ Ω we conclude ξ ∈ N Ω (x, u) as well as u ∈ T Ω (x), completing the proof.
Remark 2.1.Let F : R n → R m be continuously differentiable and let ∇F (x) denote its Jocobian.One has DF (x)(u) = ∇F (x)u and thus 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.Given f : R n → R, x ∈ dom f and a direction u ∈ R n , consider the set which we will call the analytic limiting subdiferential of f at x in direction u, following the notation from [17,Definition 1.83]. 1n this paper, inspired by directional coderivatives, we consider a direction (u, µ) ∈ R n+1 and define the limiting subdiferential of f at x in direction (u, µ) via ( The advantages of this definition are twofold: First, it leads to a finer analysis and second, there is a close relationship between subdiferentials and normal cones which allows us to easily carry over the results obtained for normal cones to subdiferentials.More detailed discussion about the two versions of directional subdiferentials is presented at the beginning of Section 4. Finally, we present some well-known properties of multifunctions as well as their directional counterparts.In order to do so, following [3], we define a directional neighborhood of (a direction) u ∈ R n .
Given a direction u ∈ R n and positive numbers ρ, δ > 0, consider the set V ρ,δ (u) given by We say that a set V is a directional neighborhood of u if there exist ρ, δ > 0 such that V ρ,δ (u) ⊂ V.Moreover, we say that a sequence x k ∈ R n converges to some x from direction u ∈ R n if for every directional neighborhood V of u we have x k ∈ x + V for sufficiently large k, or, equivalently, if there exist (t k ) ↓ 0 and u k → u with x k = x + t k u k .
Definition 2.1.Let M : R n ⇒ R m and (x, ȳ) ∈ gph M .We say that M is metrically subregular at (x, ȳ) provided there exist κ > 0 and a neighborhood U of x such that Given u ∈ R n , we say M is metrically subregular in direction u at (x, ȳ) if there exists a directional neighborhood U of u such that the above estimate holds for all x ∈ x + U.
It is well-known that metric subregularity of M at (x, ȳ) is equivalent to calmness of M −1 at (ȳ, x).We say that S : R m ⇒ R n is calm at (ȳ, x) ∈ gph S provided there exist κ > 0 and neighborhoods U of x and V of ȳ such that It is also known that neighborhood U can be reduced (if necessary) in such a way that neighborhood V can be replaced by the whole space R m , cf. [2,Exercise 3H.4].
For our purposes it is, however, suitable to employ estimate (4) without this simplification and to introduce the directional calmness by replacing V by ȳ + V, where V is the appropriate directional neighborhood.
Calmness, similarly as some other Lipschitzian stability properties enables us to estimate the images of S around (ȳ, x) via S(ȳ) and the respective calmness modulus κ.However, what we actually need for our directional calculus is the opposite, i.e., we need to be able to provide an estimate of x in terms of S(y) for y close to ȳ.This our need is reflected in the following inner version of calmness.Definition 2.2.A set-valued mapping S : R m ⇒ R n is called inner calm at (ȳ, x) ∈ gph S with respect to (w.r.t.) Ω ⊂ R m if there exist κ > 0 and a neighborhood V of ȳ such that If in the above definition V = ȳ + V, where V is a directional neighborhood of a direction v ∈ R m , we say that S possesses the inner calmness property at (ȳ, x) w.r.t.Ω in direction v.
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].
Apart from the notions of directional metric subregularity, calmness and inner calmness we will make use also of inner semicompactness and semicontinuity.
Recall that S is inner semicompact at ȳ w.r.t.Ω ⊂ R m if for every sequence y k Ω → ȳ there exists a subsequence K of N and a convergent sequence (x k ) k∈K with x k ∈ S(y k ) for k ∈ K. Given x ∈ S(ȳ), we say that S is inner semicontinuous at (ȳ, x) w.r.t.Ω ⊂ R m if for every sequence y k Ω → ȳ there exists a subsequence K of N and a sequence (x k ) k∈K with x k → x and x k ∈ S(y k ) for k ∈ K.If Ω = R m , we speak only about inner semicompactness at ȳ and inner semicontinuity at (ȳ, x).For more details we refer to [17].
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.
Clearly, inner calmness implies both inner semicontinuity and semicompactness.Interestingly, in [8, Theorem 8] Gfrerer and Outrata also investigated the estimate from definition of inner calmness and established some sufficient conditions to ensure both, calmness and inner calmness, of a class of solution maps.
Note that in case of a single-valued mapping ϕ : R m → R n , calmness and inner calmness coincide and they read as Further, ϕ is called Lipschitz continuous near ȳ in direction v if the inequality is fulfilled with V being a directional neighborhood of v.Note that Lipschitz continuity of ϕ near ȳ in direction v actually implies Lipschitz continuity of ϕ near every point y ∈ ȳ + V, y = ȳ.In construction of the directional limiting calculus one is confronted with the following issue.Given a mapping S : The task of finding an appropriate direction h is related to the following sets.Given sequences Note that exactly one of these sets is not empty, since Γ(a k , t k ) = ∅ is equivalent to t k / a k → 0.
In the situation considered above sequences a k appear in form a k ∈ S(ȳ one can clearly take a suitable direction h ∈ Γ(a k , t k ), while in the other case one can still proceed with h ∈ Γ ∞ (a k , t k ) to obtain different (but rather rough) estimates.Notation ( 6),( 7) will be extensively used throughout the whole sequel.Moreover, it is easy to see that inner calmness can be characterized in the following way.
Lemma 2.2.A set-valued mapping S : R m ⇒ R n is inner calm at (ȳ, x) ∈ gph S w.r.t.Ω in direction v if and only if for every We conclude this preparatory section with a mention concerning qualification conditions used in the calculus being developed.Analogously to [13], our main qualification condition will be the directional metric subregularity of the so-called feasibility mapping associated with the considered calculus rule.This mapping has typically the form where Ω is a closed subset of R m and ϕ : R n → R m is a continuous mapping.A tool for verifying directional metric subregularity of such mappings for continuously differentiable ϕ was recently established by Gfrerer and Klatte [7, Corollary 1] and we slightly extend this result here by allowing functions ϕ to be just calm in the prescribed direction.Proposition 2.2.Let multifunction F : R n ⇒ R m be given by F (x) = Ω − ϕ(x), where ϕ : R n → R m is continuous and Ω ⊂ R m is a closed set.Further let (x, 0) ∈ gph F and u ∈ R n be given and assume that ϕ is calm at x in direction u.Then F is metrically subregular at (x, 0) in direction u provided for all w ∈ Dϕ(x)(u) ∩ T Ω (ϕ(x)) one has the implication The proof is based on the sum rule for coderivatives of multifunctions and will be presented among other applications in Section 6.

Calculus for directional limiting normal cones
This observation allows us to consider only the indispensable directions in our estimates, as one can see in Theorems 3.1 and 3.2.
Moreover, if ϕ is calm at x in direction h we obtain a better estimate , for a fixed k and for every ε > 0 there exists a real Subregularity assumption yields existence of directional neighborhood U of h and κ > 0 such that d C (x) ≤ κd Q (ϕ(x)) holds for all x ∈ x + U and for given sufficiently large k and given ε we choose r ε such that B rε/2 (x k ) ⊂ x + U.
Next we claim that for all x ∈ B rε/2 (x k ) it holds that showing the claimed inequality.Now we consider ε k ↓ 0 and conclude that ( The fuzzy optimality conditions for problem (9), cf.[1, Theorem 2.7], [17, Lemma 2.32], state that to every η k > 0 there exist triples ( for some ξ k , ν k ∈ B. We take η k := t 2 k ↓ 0 and consider the limiting process for k → ∞.Since ( x * k + ε k )κν k is a bounded sequence, by passing to a subsequence we can assume that 6)- (7).Let us first consider the case v ∈ Γ(a k , t k ) and assume a k /t k → v.We show that z ∈ N Q (ϕ(x); v) and x * ∈ D * ϕ(x; (h, v))(z).By virtue of (11) there is a sequence of vectors z k ∈ NQ (q 2,k ) such that Since η k → 0, we obtain z k → z.Taking into account that, by virtue of the fuzzy optimality conditions, In order to show the second claim, we observe that x * k − ε k ξ k → x * and exploit in the same way as above relation (10) to show the existence of ( and hence relation ( 12) holds with q 2,k replaced by y 2,k .It follows that (y 2,k − ϕ(x))/t k → v and similarly we conclude also (x 2,k − x)/t k → h.Thus, again, since ((x, ϕ(x)) Note that in this case we have v ∈ S and t k / a k → 0 implying t k < a k for sufficiently large k.Hence, we proceed as in the previous case with t k replaced by a k and obtain the same result, the only difference being (x 2,k − x)/ a k → 0, showing the claimed relations.Observation (8) now completes the proof of the first statement.
The calmness assumption yields boundedness of (ϕ(x + t k h k ) − ϕ(x))/t k and hence we always have v ∈ Γ(a k , t k ) = ∅ and thus we only need to consider the first case.The proof is complete.
Moreover, the inner semicompactness of Ψ at ȳ w.r.t.Q in direction v is clearly implied by the assumption that An analogous assumption was used in the standard version of this result in [18,Theorem 6.43].
Then there exists i such that x + t k h k ∈ C i for infinitely many k, showing i ∈ I(x, h).Since C i ⊂ C, by passing to subsequence if necessary, we obtain

Calculus for directional limiting subdifferentials
In this section we carry over the results for normal cones from the previous section to directional limiting subdifferentials defined via normals to the epigraph by (2).However, we start by a brief discussion about the relations between the analytic directional limiting subdifferential and the one given by (2).Consider the following simple example.
In order to better understand the difference between the two concepts of directional subdifferentials, given an lsc function f : R n → R, x ∈ dom f and a direction h ∈ R n , we consider the following sets Proposition 4.1.Let f : R n → R be finite at x and lsc and consider h ∈ R n .One has On the other hand, if ν k ±∞, there exists ν such that, after passing to a subsequence if necessary, we have ν k → ν.Thus, we conclude (x * , −1) ∈ N epi f ((x, f (x)); (h, ν)) and, taking into account (x, follows directly from definition for ν = ±∞ and for ν ∈ Df (x)(h) it holds due to lsc of f and (x, f (x)) + t k (h k , ν k ) ∈ epi f .This completes the proof.
Note that one always has and this is due to the fact that for f (x k ) − f (x) and t k from definition of Note also that the calmness of an extended real-valued function is always understood with respect to its domain and hence does not exclude e.g. the indicator function of a set.Moreover, for our purposes, we in fact only need the existence of ε, κ > 0 and a directional neighborhood U of h such that suggesting that discontinuities of f also do not cause any harm.However, in order to keep the presentation as simple as possible, in the sequel we will only consider the calmness.
Corollary 4.1.Let f : R n → R be finite at x and consider h ∈ R n .Assume further that f is calm at x in direction h.Then one has This corollary shows that the results for directional limiting subdifferentials obtained later in this section can be easily carried over to analytic directional limiting subdifferentials whenever the considered function f is directionally calm.In case f fails to be calm, one can get the results for analytic directional subdifferentials using the estimate which follows from (17).
Another possible approach to calculus for directional limiting normal cones and subdifferentials would be to start with subdifferentials, build first the calculus for subdifferentials from the scratch and then carry it over to normal cones.The role of the bridge between the two concepts could be played by equivalent characterization of directional normal cones via directional subdifferentials of the indicator function or the distance function.For the sake of completeness, we present these results now.
Given a closed set C, we consider a point x ∈ C and a direction h ∈ T C (x).Clearly, δ C (•) and d C (•) are calm and directionally differentiable at x in h with δ C (x; h) = d C (x; h) = 0. Thus, taking into account Corollary 4.1, we can restrict our attention to the analytic subdifferentials.
While the relation N C (x, h) = ∂ a δ C (x; h) follows directly from definitions, in order to deal with the distance function we need to consider the following lemma.Lemma 4.1.For C, x and h as above it holds that Proof.Inclusion ⊃ follows directly from definition.Now take showing h k → h and finishing the proof.In [13], Ioffe and Outrata used subdifferentials of the distance function as the starting point for deriving the qualification conditions required for calculus rules.The previous lemma allows us to state a directional counterpart to their basic tool, [13, Proposition 3.1].
Corollary 4.3.Given C, x and h as in the previous corollary, if f : R n → R is an lsc function fulfilling f (x) = 0, ∀x ∈ C and f (x Proof.The assumptions on f imply ∂d C (x) ⊂ ∂f (x) for every x ∈ C. Hence in (19) we obtain that x * k ∈ ∂f (x + t k h k ) and the claim follows.
Finally, given f and x as before and a direction (h, ν) ∈ R n+1 , we introduce the singular subdifferential of f at x in direction (h, ν) as This notion will be used in qualification conditions which mimic their counterparts from the "standard" generalized differential calculus.As it will be shown in Corollary 5.4 below,

Chain rule and its corollaries
We start this subsection by an auxiliary result concerning separable functions, which plays a role in deriving the sum rule from the chain rule.Note that, unlike the classical case [18, Proposition 10.5], we need to impose some mild assumptions in order to obtain a reasonable estimate.Proposition 4.2 (Separable functions).Let R n be decomposed as R n = R n1 × . . .× R n l and let x = (x 1 , . . ., x l ) with x i ∈ R ni .Let f i : R ni → R be lsc for i = 1, . . ., l and let all but one of f i be calm at xi in direction h i .Set f (x) = f 1 (x 1 ) + . . .+ f l (x l ) and consider x = (x 1 , . . ., xl ) ∈ dom f and some direction (h, ν) = (h 1 , . . ., h l , ν) ∈ R n+1 .Then The proof follows from Theorem 3.2, since with ϕ : R n+l → R n+1 given by ϕ(x 1 , α 1 , . . ., x l , α l ) := (x 1 , . . ., x l , l i=1 α i ) we obtain epi f = ϕ( l i=1 epi f i ).Moreover, it can be shown that is inner calm at ((x, f (x)), (x 1 , f 1 (x 1 ), . . ., xl , f l (x l ))) w.r.t.epi f in direction (h, ν), due to the calmness of all but one of f i , even when the calmness is considered in the sense of (18).For the sake of brevity, the technical details are skipped.
Remark 4.1.Clearly, the calmness of all but one f i is just a sufficient condition that can be replaced by requiring the inner calmness of mapping (22).Moreover, one can also apply Theorem 3.2 without these assumptions to obtain more complicated estimates.
We have also stated the result concerning singular subdifferentials (21), because we will need it for deriving qualification conditions for the sum rule.Later on we will not write down the results for singular subdifferentials although usually the proofs will be applicable to this case as well.
Since the calmness of ϕ at x in h is equivalent to the calmness of Ψ at (x, f (x)) in direction (h, ν), we obtain only the first possibility and hence the appropriate simpler estimate.This finishes the proof.
Again, taking into account Propositions 2.2 and 4.2, the metric subregularity of F from Corollary 4.5 is implied by the condition Inclusion ( 23) holds true in particular if all but one of f i are Lipschitz continuous near x in direction h.Naturally, Remark 4.1 applies here as well.

Directional limiting subdifferentials of special functions
We conclude this section with estimates for directional limiting subdifferentials of the pointwise maximum and minimum of a finite family of functions, the distance function and the value function.
Given a function g and a point x ∈ dom g, one always has that (y * , −β) ∈ Nepi g (x, g(x)) implies β ≥ 0 and hence we obtain β i ≥ 0 for all i ∈ I(x, (h, ν)).
, showing also J ⊂ I 0 (x, (h, ν)).On the other hand, for i / ∈ J we have β i = 0 and hence ).This completes the proof.
Taking into account the differentiability of ϕ, Theorem 3.2 now yields all statements of the theorem.

Calculus for directional coderivatives
In the first part of this section we present two basic calculus rules, namely the chain rule and the sum rule for directional limiting coderivatives.In fact, having proved one of them, the other one can be derived relatively easily on the basis of the first one, similarly like in the case of standard limiting coderivatives.Here we follow essentially the pattern from [18].Thereafter we present a "scalarization" formula which may facilitate the computation of coderivatives of single-valued Lipschitz continuous mappings.
To compute an estimate of N gph S ((x, ū); (h, l)), we invoke first Theorem 3.2, which is possible thanks to condition (a), see also Remark 3.1.We obtain that Next we compute N C ((x, w, ū); (h, k, l)) via Theorem 3.1.Thanks to condition (b) and the calmness of H one has and, likewise, Further we observe that so that the first union in (27) with respect to k can be taken over the set and the second union in (27) with respect to k can be taken over the set Using consecutively representations (30), (31) and inclusions (28), ( 29) we obtain that It follows that for u * := −y * 2 one has and the proof is complete.
Let us comment on assumption (b) which is, admittedly, not easy to verify in general.Following Proposition 2.1, it may be ensured by the next two conditions (which are, however, more restrictive).
(i) For all w ∈ Ξ(x, ū) and all directions k such that k ∈ DS (ii) for all w ∈ Ξ(x, ū) and all directions k ∈ S such that k We observe that both conditions (32), (33) are automatically fulfilled provided either S 1 is metrically regular around (x, w) for w ∈ Ξ(x, ū), or S 2 has the Aubin property around ( w, ū) for w ∈ Ξ(x, ū).This complies with the corresponding conditions in [18,Theorem 10.37].More precisely, one can employ the characterizations of the directional metric regularity and the Aubin property from [7, Theorem 1].Further note that in inclusion (26) the union over k ∈ {ξ ∈ S|ξ ∈ DS 1 (x, w)(0), 0 ∈ DS 2 ( w, ū)(ξ)} vanishes provided we strengthen assumption (a) by asking that the intermediate mapping Ξ is inner calm at (x, ū, w) (w.r.t.gph S) in direction (h, l).
On the basis of the above considerations we obtain immediately the following corollaries of Theorem 5.1.
Note that the single-valuedness and the Lipschitz continuity of S 1 are carried over to the intermediate mapping Ξ, yielding the fulfillment of assumption (a) as well as the reduction in the estimate.It is not difficult to verify that the properties of S 1 enable us to simplify the multifunction (25) by replacing its first row by S 1 (x)−w.We make use of this fact in the formulation of Theorem 5.2.
Corollary 5.2.In the framework of Theorem 5.1 let S 2 be single-valued and Lipschitz continuous near every w ∈ S 1 (x).Further, let assumption (a) be fulfilled.Then inclusion (26) with evident simplifications holds true.
The assumptions imposed on S 2 justify assumption (b), since for a single-valued mapping Lipschitz continuity and the Aubin property coincide.
Corollaries 5.1 and 5.2 represent our main tool in the proof of the next statement.We consider there mappings S i : R n ⇒ R m , i = 1, 2, . . ., p, and associate with them the multifunction Ξ : R n × R m ⇒ (R m ) p defined by Ξ(x, u) = {w = (w 1 , w 2 , . . ., w p ) ∈ (R m ) p |w i ∈ S i (x), w i = u}.
Corollary 5.3.In the setting of Theorem 5.2 assume that p = 2 and S 1 is single-valued and Lipschitz continuous near x and directionally differentiable at x. Then all assumptions of Theorem 5.2 are fulfilled and where k = S 1 (x; h).
Note that inclusion (37) becomes equality provided S 1 is continuously differentiable near x, cf.
An equivalent formulation of the following useful result was also proven in [16, Corollary 5.9].

Applications
In this section we apply some of the above presented calculus rules to several problems of variational analysis, where directional notions can be advantageously utilized.

First-order sufficient conditions for directional metric regularity and subregularity of feasibility mappings
Consider a mapping of the form F (x) = Ω−ϕ(x) which arises in qualification conditions throughout the whole paper.The next result (announced already in Section 2) extends the results from [7, Theorem 1, Corollary 1], where ϕ is assumed to be continuously differentiable.
Theorem 6.1.Let the multifunction F : R n ⇒ R m be given by F given and assume that ϕ is calm at x in direction u.Then 1. F is metrically subregular at (x, 0) in direction u provided for all w ∈ Dϕ(x)(u) ∩ T Ω (ϕ(x)) one has the implication The proof of the second statement is based on [3, Theorem 5] which provides equivalent characterizations of directional metric regularity and in finite dimensional spaces one of them states that F is metrically regular at (x, 0) in direction (u, v) if and only if 0 ∈ D * F ((x, 0); (u, v))(λ) implies λ = 0.The first statement then follows from the fact that metric regularity of F at (x, 0) in direction (u, 0) implies metric subregularity of F at (x, 0) in direction u.Thus, it suffices to show the following lemma.Lemma 6.1.Under the settings and assumptions of the previous theorem we have Proof.The assumed calmness of ϕ implies that the intermediate mapping Ξ(x, y) = {y+ϕ(x), −ϕ(x)} is inner calm at (x, 0, ϕ(x), −ϕ(x)) in direction (u, v).On the other hand, denoting by G the mapping that to each x assigns set Ω, it is clear that G has the Aubin property and we may apply the sum rule for coderivatives, Theorem 5.2.The statement of the lemma thus follows from the fact that for some w we obtain

Subtransversality of set systems
Then collection {C 1 , C 2 , . . ., C l } is subtransversal at x. Very often the sets C i correspond to various constraint systems and can be described as with Q i ⊂ R mi being closed and ϕ i : R n → R m being Lipschitz continuous near x.As a simple consequence of Theorem 6.2 we obtain a condition ensuring the subtransversality of a collection of pre-images.
Corollary 6.1.In the setting of Theorem 6.2 assume that the sets C i are given via (38) where, in addition to the posed assumptions, functions ϕ i are directionally differentiable at x.Further assume that the mappings are metrically subregular at x. Finally suppose that there do not exist nonzero vectors u Then collection {C 1 , C 2 , . . ., C l } is subtransversal at x.
The proof follows easily from Theorem 6.2, Theorem 3.1 and the fact that the set on the right-hand side of (39) amounts exactly to and x = (0, 0).It is easy to verify that all assumptions of Corollary 6.1 are fulfilled and the only direction u satisfying (39) is the direction R + (1, 0).Clearly, with ϕ 1 and Q 1 given in (42) one has Consequently, there does not exist any nonzero pair v * 1 , v * 2 satisfying conditions (40), (41) and so collection {C 1 , C 2 } is subtransversal at x.
Note that we are not able to detect this property via the (stronger) Aubin property of S because, with p = 0, Thus, since the vectors a * = (−1, 0) and b * = (1, 0) belong to D * S(p, x)(0), we conclude from the Mordukhovich criterion that S does not possess the Aubin property around (p, x).(ii) M is metrically subregular at (p, x, 0);
Then S has the Aubin property around (p, x).This statement remains valid if conditions (ii), (iii) are replaced by the (stronger) implication In the second case one has {1} ∈ I 0H (p, x) and so we have to verify the conditions y * > 0 : Since both these conditions are fulfilled, the corresponding mapping S has the Aubin property around (p, x).
Note that the standard condition ensuring the Aubin property of S via the Mordukhovich criterion is violated.Indeed, by the calculus from [17, Chapter 3] one has One can easily verify that, for instance, the nonzero pair (q * , y * ) = (0.5, −1) satisfies the system q * 0 ∈ ∂ y * , M (p, x) and so the estimate (45) is not precise enough to enable us to detect the Aubin property of S around (p, x) via the Mordukhovich criterion.

Improving the standard calculus
It can easily be seen that all rules presented in Sections 3-5 reduce to their counterparts from the classical generalized differential calculus provided we set the considered directions to be zero.In some cases, however, the classical rules may even be improved when one employs the appropriate results from this paper.This concerns both the restrictiveness of the imposed assumptions as well as the sharpness of the resulting estimates.
As to the former case, Proposition 6.1 below extends a statement from [18,Theorem 6.43] by a substantial relaxation of the assumptions.Proposition 6.1.Consider a closed set C ⊂ R n and a continuous mapping ϕ : R n → R l , set Q = ϕ(C) and consider ȳ ∈ Q.Let Ψ : R l ⇒ R n be given by Ψ(y) := ϕ −1 (y) ∩ C and let it be inner semicontinuous at (ȳ, x) w.r.t.Q for some x ∈ Ψ(ȳ).Assume further that the set-valued mapping F : R l+n ⇒ R 2(l+n) given by F (y, x) = gph ϕ −1 − (y, x) × (R l × C) − (y, x) is metrically subregular at ((ȳ, x), (0, . . ., 0)).Then fact that in this development one needs a "regular" behavior of feasibility mappings only in the relevant directions.In the rules relying (partially) on Theorem 3.2 we make use of a special "inner calmness" property which arose (under a different name) also in completely different contexts.In Section 6 we collected some representative (classes of) problems, where directional limiting objects are helpful and the results of this paper enable the user to compute them.

dTheorem 6 . 2 .
Consider the collection of closed sets C 1 , C 2 , . . ., C l from R n and a point x ∈ C := l i=1 C i .By the definition (cf., e.g., [15, Definition 1(ii)]), these sets are subtransversal at x provided there exist a neighborhood U of x and a constant L > 0 such that the metric inequalityd C (x) ≤ L l i=1 Ci (x)holds for all x ∈ U .This is, on the other hand, equivalent with the calmness of the perturbation mapping S(p 1 , . . ., p l ) = {x|p i + x ∈ C i , i = 1, 2, . . ., l} at (0, . . ., 0, x), cf.[13, Section 3].A straightforward application of [8, Theorem 3.8] yields the following result.Assume that there do not exist nonzero vectors u

By combination of [ 8 ,
Theorem 4.4] with Proposition 5.1 one obtains a sufficient condition for the Aubin property for a class of implicitly defined multifunctions.Let the function M : R l ×R n → R m be Lipschitz continuous near the reference point (p, x) ∈ R l × R n satisfying M (p, x) = 0 and consider the solution mapping S(p) := {x ∈ R n |M (p, x) = 0}.Theorem 6.3.Assume that M is directionally differentiable at (p, x) and (i) {u ∈ R n |M ((p, x); (v, u)) = 0} = ∅ for all v ∈ R l ;