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Lipschitz-like property relative to a set and the generalized Mordukhovich criterion

In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a closed and convex set by virtue of the projection of the coderivative onto a tangent cone. Furthermore, by introducing a projectional coderivative of set-valued mappings, we establish a verifiable generalized Mordukhovich criterion for the Lipschitz-like property relative to a closed and convex set. We will study the representation of the graphical modulus of a set-valued mapping relative to a closed and convex set by using the outer norm of the corresponding projectional coderivative value. For an extended real-valued function, we will apply the obtained results to investigate its Lipschitz continuity relative to a closed and convex set and the Lipschitz-like property of a level-set mapping relative to a half line.


Introduction
Stability theory of set-valued mappings has been extensively investigated. The monographs [5,9,[24][25][26] contain a comprehensive presentation for the derivation of conditions ensuring various stability properties of set-valued mappings, including the Lipschitz-like property.
The Lipschitz-like property of set-valued mappings (known also as Aubin property, or pseudo-Lipschitz property) has been introduced in [3]. The Lipschitz-like property has been well studied in the literature by virtue of Mordukhovich criterion, which was initially developed by [22] and a more direct proof was given in [26] by using the basic variational analysis tools. One important assumption when Mordukhovich criterion is applied is that the candidate parameter under consideration is in the interior of the domain of set-valued mappings. It is worth noting that this criterion has been applied in the study of the solution mapping of generalized equations, linear semi-infinite and infinite systems and a parametric linear constraint system in [6,14,20] respectively. The Lipschitz-like property of the stationary set of some constrained optimization problems has been explored in [19,20]. A critical face condition was developed in [8] as necessary and sufficient conditions for the Lipschitz-like property of the solution mapping of a linear variational inequality problem and a nonlinear variational inequality problem (via linearization) over a polyhedral set.
Metric regularity relative to a set was studied in [2,15,16] by using strong slopes, and directional metric (sub)regularity, isolated calmness and Lipschitz-like property relative to a set were explored in [4,12] by using directional limiting coderivative respectively.
In this paper we will study the Lipschitz-like property relative to a set for set-valued mappings. When the restricted set is closed, we will obtain some necessary condition and sufficient condition respectively for set-valued mappings to have the Lipschitz-like property relative to the set by employing regular normal cone and limiting normal cone of a restricted graph of the setvalued mapping and the tangent cone of the set near the candidate point. When the set is closed and convex, we will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to the set by virtue of the projection of the coderivative onto the tangent cone. Furthermore, by employing an outer limit of the projection of the coderivative onto the tangent cone, we will introduce a projectional coderivative of set-valued mappings and apply it to establish a verifiable generalized Mordukhovich criterion for the Lipschitz-like property relative to the set. We will study the representation of the graphical modulus of a set-valued mapping relative to a closed and convex set by using the outer norm of the corresponding projectional coderivative value. When the set is merely closed, we will compare our sufficient condition with the one that is derived via the directional limiting coderivative in [4].
For an extended real-valued function, the Lipschitz continuity relative to a closed set is equivalent to that its profile mapping has Lipschitz-like property relative to the set. By virtue of this equivalence, we will apply the obtained results to derive a full characterization for an extended real-valued function to have the Lipschitz continuity relative to a closed and convex set of the domain, including that of a sublinear function. Moreover, a reinterpretation of subgradients of an extended real-valued function f has been pointed out in [26,Theorem 9.41] by applying the Mordukhovich criterion. That is, level-set mapping S : α → {x| f (x) − v, x −x ≤ α} fails to have the Lipschitz-like property if and only ifv ∈ ∂ f (x). Whenever f is convex andv ∈ ∂ f (x), x is clearly a minimum of f (x) − v, x −x implying that domS = {α|α ≥ f (x)} and hence that f (x) int(domS ). By virtue of the generalized Mordukhovich criterion, we will show that S fails to have the Lipschitz-like property relative to domS at f (x) forx if and only ifv belongs to ∂ > v f (x), the outer limiting subdifferential set of f atx with respect tov, see its definition in Section 4. In the case ofv = 0, ∂ > v f (x) reduces to the outer limiting subdifferential set ∂ > f (x), which has been studied extensively for the study of error bounds in the literature, see [7,10,11,[16][17][18]21].
The organization of the paper is as follows. In Section 2, we will employ regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping to obtain some neighborhood necessary condition and sufficient condition respectively for set-valued mappings to have the Lipschitz-like property relative to a closed set. We will introduce a projectional coderivative of set-valued mappings and apply it to establish a verifiable generalized Mordukhovich criterion for the Lipschitz-like property relative to a closed and convex set. In Section 3, we will obtain characterization of the Lipschitz continuity relative to a closed and convex set of an extended realvalued function. In Section 4, we will apply the obtained results to investigate the Lipschitz-like property relative to the half line for a level-set mapping.
Throughout the paper we use the standard notations of variational analysis; see the seminal book [26] by Rockafellar and Wets. For readers' convenience, we also mention alternative names for some of notions used in this paper, see [24]. The Euclidean norm of a vector x is denoted by ||x||, and the inner product of vectors x and y is denoted by x, y . We denote by [x] ⊥ := {v ∈ Ê n | v, x = 0} the orthogonal space of the vector x. Let B denote the closed unit Euclidean ball and let S denote the unit sphere. We denote by B δ (x) the closed ball centered at x with radius δ > 0.
Let A ⊂ Ê n be a nonempty set. We say that A is locally closed at a point x ∈ A if A∩U is closed for some closed neighborhood U of x. We denote the interior, the relative interior, the closure, the boundary, the convex hull and the positive hull of A respectively by int A, ri A, cl A, bdry A, conv A and posA := {0} ∪ {λx|x ∈ A and λ > 0}. We denote by A ⊥ := {v ∈ Ê n | v, x = 0 ∀x ∈ A} the orthogonal space of A, and by the horizon cone of A. The polar cone of A is defined by The support function σ A of A is defined by The indicator function δ A of A is defined by The projection mapping proj A is defined by For a set M ⊂ Ê n , we denote the projections of M on A by Let x ∈ A. We use T A (x) to denote the tangent/contingent cone to A at x, i.e. w ∈ T A (x) if there exist sequences t k ↓ 0 and {w k } ⊂ Ê n with w k → w and x + t k w k ∈ A ∀k. We denote by A is said to be regular at x in the sense of Clarke if it is locally closed at x and N A (x) = N A (x).
Let C ⊂ Ê n be a nonempty convex set. A face of C is a convex subset C ′ of C such that every closed line segment in C with a relative interior point in C ′ has both endpoints in C ′ . An exposed face of C is the intersection of C and a non-trivial supporting hyperplane to C. In other words, F is an exposed face of C if and only if there is some x ∈ Ê n such that F = arg max v∈C x, v . See the book [27] for more details. or in other words, gph S is a cone. If S is a positively homogeneous mapping, the outer norm of S is denoted and defined by |S | + := sup which is the infimum over all constants κ ≥ 0 such that u ≤ κ x for all pairs (x, u) ∈ gph S .
Consider a pointx ∈ dom S . The outer limit of S atx is defined by S is said to be outer semicontinuous atx if The (normal) coderivative, the regular/Fréchet coderivative and the proximal coderivative of S at x for anyū ∈ S (x) are respectively the mapping D * S (x |ū) : Ê m ⇒ Ê n defined by and the mapping D * prox S (x |ū) : Ê m ⇒ Ê n defined by Clearly, the following inclusions hold: For a set X ⊂ Ê n , we denote by the restricted mapping of S on X. It is clear to see that Let g : Ê n → Ê := Ê ∪ {±∞} be an extended real-valued function and letx be a point with g(x) finite. We denote the epigraph and the domain of g by respectively. g is said to be locally lower semicontinuous (for short, lsc) atx, if there is an ǫ > 0 such that all sets of the form {x | x −x ≤ ǫ, g(x) ≤ α} with α ≤ g(x) + ǫ are closed, see [26,Definition 1.33]. It is well-known that g is locally lsc atx if and only if epi g is locally closed at (x, g(x)).
The vector v ∈ Ê n is a regular/Fréchet subgradient of g atx, written v ∈ ∂g(x), if The vector v ∈ Ê n is a (general/basic) subgradient of g atx, written v ∈ ∂g(x), if there exist and v k ∈ ∂g(x k ). The subdifferential set ∂g(x) is also referred to as limiting/Mordukhovich subdifferential. The vector v ∈ Ê n is a horizon/singular The outer limiting subdifferential of g atx introduced in [17] is denoted and defined as follows: which coincides with the outer limiting subdifferential set ∂ > 0 g(x) of g atx with respect to 0, a notion newly introduced in Section 4. Note that a closely related notion, called the right-sided subdifferential, was given in [23] and defined by using a weak inequality instead of the strict inequality used here. See [24, Definition 1.100 and Theorem 1.101] for more details on the rightsided subdifferential and its applications.

The general case
In this section, we consider a set-valued mapping S : Ê n ⇒ Ê m , a pair (x,ū) ∈ gph S and a set X ⊂ Ê n withx ∈ X. First, we formally give the definition for the Lipschitz-like property of S relative to X atx forū and accordingly the definition for the graphical modulus lip X S (x |ū). In the case of X being closed, by using the (regular) coderivative of the restricted mapping S | X , we then present a (necessary) sufficient condition for the relative Lipschitz-like property. Finally by filling the gap between these necessary and sufficient conditions, we will focus on the case that X is closed and convex. We give two characterizations for the relative Lipschitz-like property and two corresponding formulas for the graphical modulus lip X S (x |ū) when X is closed and convex: one is a uniform boundedness condition getting all nearby points involved, and the other one is a point-based condition based on the so-called projectional coderivative, a newly introduced notion. In both of these characterizations, the projections of the coderivative of the restricted mapping S | X onto the tangent cones to X play a key role.
The definition below is borrowed from [26,Definition 9.36].
Definition 2.1 (Lipschitz-like property relative to a set). A mapping S : Ê n ⇒ Ê m has the Lipschitz-like property relative to X atx forū, wherex ∈ X andū ∈ S (x), if gph S is locally closed at (x,ū) and there are neighborhoods V ∈ N(x), W ∈ N(ū), and a constant κ ∈ Ê + such The graphical modulus of S relative to X atx forū is then In the case of X being closed, we first present a necessary condition for the Lipschitz-like property relative to X by using the tangent cone T X (x) and the regular coderivatives D * S | X (x | u) of S | X for all nearby points (x, u) of (x,ū) in gph S | X . Theorem 2.1 (Necessity). Consider S : Ê n ⇒ Ê m ,x ∈ X ⊂ Ê n ,ū ∈ S (x) and κ ≥ 0. Suppose that X is closed. If S has the Lipschitz-like property relative to X atx forū with constant κ, then the condition max holds for all (x, u) close enough to (x,ū) in gph S | X .
Proof. As S has the Lipschitz-like property relative to X atx forū with constant κ, there exist some neighborhoods V ∈ N(x) and W ∈ N(ū) such that Without loss of generality, we can assume that the sets V and W are open. Let x * ∈ D * S | X (x | u)(u * ) (i.e., (x * , −u * ) ∈ N gph S ∩(X×Ê m ) (x, u)) with (x, u) ∈ gph S | X ∩ (V × W), and let w ∈ T X (x) ∩ S. By the definition of tangent cone, there exists some {x k } ⊂ X\{x} such that x k → x and Clearly, there is some k 0 such that x k ∈ X ∩ V for all k ≥ k 0 . This, together with the facts that x ∈ X ∩ V and u ∈ S (x) ∩ W, implies by (3) the existence of u k ∈ S (x k ) such that Clearly, we have u k → u. In view of (x * , −u * ) ∈ N gph S | X (x, u), we get from the definition of regular normal cone (cf. [26, Defition 6.3 or 6(5)]) that By (5), we have which together with (4) and (6) implies the inequality x * , w ≤ κ u * and hence (2). This completes the proof.
In the case of X being closed, we now present a sufficient condition for the Lipschitz-like property relative to X by using the closure of the generated cone cl pos(X−x) and the coderivatives D * S | X (x | u) of S | X for all nearby points (x, u) of (x,ū) in gph S | X . In our proof, the Ekeland's variational principle plays a key role.
Suppose that gph S is locally closed at (x,ū) and that X is closed. If the condition holds for all (x, u) close enough to (x,ū) in gph S | X , then S has the Lipschitz-like property relative to X atx forū with constantκ.
Proof. Observing that all the properties involved depend on the nature of gph S in an arbitrary small neighborhood of (x,ū), there's no harm, therefore, in assuming from now on that gph S is closed in its entirety.
Let 0 < ε <˜κ −κ 4κ . Suppose by contradiction that S does not have the Lipschitz-like property relative to X atx forū with constantκ, meaning that there exist Clearly, we have 0 < β ≤ 2κε.
Clearly, ϕ is lsc (due to closedness of gph S and X) with inf ϕ being finite, and By equipping the product space Ê n × Ê m with a norm p defined by we apply the Ekeland's variational principle to obtain some (x,ũ) ∈ Ê n × Ê m such that arg min From (10), it follows that and hence that Then by the triangle inequality, we have From (9), it follows that and hence by the triangle inequality that So we havex x ′ , for otherwise we have contradicting to (8). From (11) where Clearly, ψ is convex and Lipschitz continuous and in terms of closed unit balls B 1 in Ê n and B 2 in Ê m , Applying the calculus rule for subgradients of Lipschitzian sums [26, Exercise 10.10], we deduce from (15) that This, together with (16), implies the existence of such that Sincex, x ′ ∈ X withx x ′ , we have Then we have where the first inequality follows from the Cauchy-Schwarz inequality, the second one from the fact that β ≤ 2κε, and the last one from our setting that ε <˜κ −κ 4κ . Therefore, we have In view of (12)(13)(14), (17)(18) and the fact that ε could be any number such that 0 < ε <˜κ −κ 4κ , we conclude that condition (7) cannot hold for all (x, u) close enough to (x,ū) in gph S | X , a contradiction. This completes the proof.
Whenever X is not only closed but also convex, the gap between the previous necessary and sufficient conditions will be filled, and even a formula for the graphical modulus lip X S (x |ū) can be provided. Theorem 2.3 (Lipschitz-like property relative to a closed and convex set). Consider S : Ê n ⇒ Ê m ,x ∈ X ⊂ Ê n andū ∈ S (x). Suppose that gph S is locally closed at (x,ū) and that X is closed and convex. The following properties are equivalent: (a) S has the Lipschitz-like property relative to X atx forū.
(b) There is some κ ≥ 0 such that the condition holds for all (x, u) close enough to (x,ū) in gph S | X .

Moreover, we have
Alternatively, the coderivative D * S | X (x | u) in (19) as well as in (20) can be equivalently replaced by the regular coderivative Proof. To show the equivalent replacement, assume that the following inequality holds for all (x, u) close enough to (x,ū) in gph S | X : Let (x, u) be close enough to (x,ū) in gph S | X and let . It then follows from (21) that for all k large enough, where the equality follows from the projection theorem [26, Exercise 12.22] for a closed and convex cone and its polar. So we have for all k large enough, Let w ∈ T X (x) ∩ S be given arbitrarily. Since X is closed and convex (implying that w is a regular tangent vector to X at x), it follows from regular tangent cone properties [26, Theorem 6.26] that there exists some w k ∈ T X (x k ) such that w k → w. So we have for all k large enough, where the equality follows also from the projection theorem [26,Exercise 12.22]. So starting from (21), we assert that the following condition holds for all (x, u) close enough to (x,ū) in gph S | X : This, together with the inclusions indicates that the coderivative D * S | X (x | u) in (19) as well as in (20) can be equivalently replaced by the regular coderivative In what follows, let [(a) =⇒ (b)] Assuming (a), we will show (b) by proving the inequality Choose any κ ∈ (lip X S (x |ū), +∞). Then S has the Lipschitz-like property relative to X atx forū with constant κ. It then follows from Theorem 2.1 that the following condition holds for all (x, u) close enough to (x,ū) in gph S | X : By the same argument used earlier and the equivalent replacement as we have already shown, we assert that the following condition holds for all (x, u) close enough to (x,ū) in gph S | X : which implies (b) and hence the inequality β ≤ κ. So the inequality (22) follows.
[(b) =⇒ (a)] Assuming (b), we will show (a) by proving the inequality from which the equality (20) follows as the inequality in the other direction has been proved earlier. Suppose by contradiction that the inequality (24) does not hold. Choose any κ ′ such that β < κ ′ < lip X S (x |ū). Clearly, S fails to have the Lipschitz-like property relative to X atx forū with constant κ ′ . In view of the fact that T X (x) = cl pos(X − x) for all x ∈ X due to X being closed and convex, we deduce from Theorem 2.2 that the inequality cannot be fulfilled for all (x, u) close enough to (x,ū) in gph S | X and (x * , −u * ) ∈ N gph S | X (x, u). By the same argument used earlier, the inequality (25) amounts to Clearly, we have w k > 0 for all k. Then we have for all k, contradicting to the assumption that β < κ ′ . This completes the proof.
Motivated from Theorem 2.3, we can provide a point-based criterion for the relative Lipschitzlike property via the projectional coderivative defined below by first projecting the coderivative for all nearby points onto the tangent cones and then taking the outer limits for the projections.
Definition 2.2 (Projectional coderivatives). Consider a mapping S : Ê n ⇒ Ê m and a point x ∈ X ⊂ Ê n . The projectional coderivative of S atx for anyū ∈ S (x) with respect to X is the Theorem 2.4 (generalized Mordukhovich criterion). Consider S : Ê n ⇒ Ê m ,x ∈ X ⊂ Ê n and u ∈ S (x). Suppose that gph S is locally closed at (x,ū) and that X is closed and convex. The following properties are equivalent: (a) S has the Lipschitz-like property relative to X atx forū.
Proof. It is clear to see from the definition of projectional coderivatives that, the mapping D * X S (x | u) is outer semicontinuous and positively homogeneous. Then the equivalence of (b) and (c) follows immediately from [26,Proposition 9.23]. The equivalence of (a) and (c), and the formula for lip X S (x |ū) can be proved in a similar way as in the proof of Theorem 2.3. The detailed proof is omitted. This completes the proof.
The important role played by the projectional coderivative D * X S (x |ū) in the study of the relative Lipschitz-like property, is revealed by the generalized Mordukhovich criterion above. In the following remarks, we list some simple facts on the projectional coderivatives.
Remark 2.1. In the case ofx ∈ int X, we have for allū ∈ S (x), where ker B := {x ∈ Ê n | Bx = 0}. While in the case of a closed half-space we have for allx ∈ bdry X, Let (p,x) ∈ gph S and let F (p,x) be the collection of the faces of K that contain Ax +p. Then we have where N F := N K (Ax + p) for any F ∈ F (p,x) and any choice of (p, x) such that Ax + p ∈ ri F. To show (28), we rely on the definition of the projectional coderivative by combining the following facts: (i) F (p,x) consists of finitely many faces of K; (ii) for any face F of K, T K (Ax + b) and N K (Ax + b) are both constants whenever Ax + p ∈ ri F; (iii) for any sequence (p k , x k ) → (p,x) with (p k , x k ) ∈ gph S for all k, there exists some F ∈ F (p,x) such that, by taking a subsequence if necessary, Ax k + p k ∈ ri F for all k; (iv) for any F ∈ F (p,x) and any (p, x) with Ax + p ∈ ri F, In contrast, by the definition of the coderivative, we have The classical Mordukhovich criterion D * S (p |x)(0) = {0} amounts to While the generalized Mordukhovich criterion D * dom S S (p |x)(0) = {0} holds automatically as it amounts to the following trivial equalities: It is interesting to note that meaning that the projectional coderivative D * dom S S (p |x) and the coderivative D * S (p |x) differs from each other only whenp is on the boundary of dom S .
As suggested by one reviewer, we will compare our sufficient condition with the one established for an implicitly defined set-valued mapping in terms of a directional limiting coderivative in [4,Theorem 3.5]. To have a better comparison, we first present an explicit version of [4, Theorem 3.5] as follows.
We start by recalling the definitions of the directional limiting normal cone and the directional limiting coderivative. For a set Ω ⊂ Ê n withx ∈ Ω and a direction u ∈ Ê n , the directional limiting normal cone to Ω in direction u atx is defined by while for a set-valued mapping S : Ê n ⇒ Ê m having locally closed graph around (x,ȳ) ∈ gph S and a pair of directions (u, v) ∈ Ê n × Ê m , the set-valued mapping D * S ((x,ȳ); (u, v)) : Ê m ⇒ Ê n , defined by is called the directional limiting coderivative of S in the direction (u, v) at (x,ȳ). See [4,13] and references therein for more details and some basic properties of these notions.
Theorem 2.5 ( [4, Theorem 3.5] in an explicit form). Consider S : Ê n ⇒ Ê m ,x ∈ X ⊂ Ê n and u ∈ S (x). Assume that gph S is locally closed at (x,ū) and that X is closed. Further assume that the following conditions are satisfied: (i) For every x ∈ T X (x) and every sequence t k ↓ 0, there exists some u ∈ Ê n such that This holds in particular if, for every x ∈ T X (x), there is some u ∈ Ê m such that (x, u) ∈ holds for all x ∈ T X (x) and (x, u) ∈ T gph S (x,ū) with (x, u) (0, 0).
Then S has the Lipschitz-like property relative to X atx forū.
It is clear to see that gph S is locally closed at (x,ū) if and only if gph M is locally closed at (x,ū, 0), and that condition (i) holds if and only if, for every x ∈ T X (x) and every sequence t k ↓ 0, there exists some u ∈ Ê m such that Moreover, by definition, we have the following equivalences:

and 2.5 for the Lipschitz-like property of S relative to X atx for u. Unlike Theorem 2.2, which utilizes integrated information behind the coderivative of the restriction mapping S | X (combining the local behavior of S around (x,ū) and of X aroundx as a whole), Theorem 2.5 treats S and X as 'separated variables' (the local behavior of S around (x,ū) described by the directional limiting coderivative D * S ((x,ū); (·, ·)) is independent of the local behavior of X aroundx). So it would be the case that the sufficient condition in Theorem 2.5 is easier to be verified than that in Theorem 2.2. However, in the case of X being not only closed but also convex, the sufficient condition in Theorem 2.turns out to be also necessary as can be seen from Theorems 2.3 and 2.4, but the sufficient condition in Theorem 2.5 is far from being necessary as will be seen from Example 2.1 below.
To end this section, we demonstrate by an interesting example how our results in Theorems 2.3 and 2.4 can be applied in the circumstance that the graph can be decomposed into finitely many simple pieces. Moreover, by this example, we also demonstrate how Theorem 2.5 can fail in identifying the relative Lipschitz-like property.

Clearly, we have
In terms of Then for every (q, x) ∈ gph S , there is a unique (I 1 , I 2 , I 3 ) ∈ I such that (q, x) ∈ (gph S ) (I 1 ,I 2 ,I 3 ) and where Ω : This implies by definition that So by the Mordukhovich criterion [26,Theorem 9.40], we assert that S does not have the Lipschitzlike property atq forx, which can also be seen from the fact thatq ∈ bdry dom S .
In what follows, we will apply Theorem 2.3 to study the Lipschitz-like property of S relative to dom S atq forx, which amounts to the existence of some κ ≥ 0 such that holds for all (q, x) close to (q,x) in gph S , or in other words, for all (q, x) close to (q,x) in all (gph S ) (I 1 ,I 2 ,I 3 ) with (I 1 , I 2 , I 3 ) ∈ I. For each (I 1 , I 2 , I 3 ) ∈ I, we define κ(I 1 , I 2 , I 3 ) := inf κ ≥ 0 | (31) holds for all (q, x) ∈ (gph S ) (I 1 ,I 2 ,I 3 ) .
Then by some direct calculation, we have For instance, whenever (I 1 , I 2 , For another instance, whenever (I 1 , I 2 , and κ(I 1 , I 2 , In view of (32), we get from Theorem 2.3 that and hence that S does have the Lipschitz-like property relative to dom S atq forx. In contrast, we can also apply Theorem 2.4 to obtain the same result by calculating the projectional coderivative D * dom S S (q |x), which can be done in the same way by decomposing gph S into finitely many pieces (gph S ) (I 1 ,I 2 ,I 3 ) . The details are complicated and thus omitted.
Let Q be a closed subset of dom S such thatq ∈ Q and q := (0, −1) T ∈ T Q (q) (e.g., dom S could be the largest instance of Q). Our argument above suggests that S has the Lipschitz-like property relative to Q atq forx, which, however, cannot be verified via Theorem 2.5. To this end, we argue that the condition D * S ((q,x); (q, x))(0) = {0} cannot be fulfilled when x := (0, 1) T . By definition, we have in terms of (I 1 , In view of the facts that gph S is the union of finitely many convex polyhedral cones and that (q,x) = (0, 0), we have (q, x) ∈ T gph S (q,x) = gph S and N gph S ((q,x); (q, x)) := lim sup t↓0, where the second equality follows from the fact that gph S is a closed cone, the third one from the outer semi-continuity of the normal cone mappings, and the last one from (30). Then we have by definition, suggesting that the Lipschitz-like property of S relative to Q atq forx cannot be derived from Theorem 2.5.

Profile mappings and relative Lipschitzian continuity
Consider a function f : Ê n → Ê, a pointx ∈ Ê n where f is finite and locally lsc, and a set X ⊂ dom f such thatx ∈ X. The notion of relative Lipschitzian continuity of f is standard, see [26, Definition 9.1 (b)] for a formal definition. To say that f is locally Lipschitz continuous atx relative to X is to assert the following inequality: It is straightforward to verify that f is locally Lipschitz continuous atx relative to X if and only if the profile mapping has the Lipschitz-like property relative to X atx for f (x), and furthermore that their moduli are equal: In what follows, we will study the relative Lipschitzian continuity by applying Theorem 2.3 to the profile mapping E f , and will give subgradient characterizations for the relative Lipschitzian continuity.
The classical local Lipschitzian continuity of f atx (without mention of X) has been fully characterized by virtue of the (horizon) subgradients of f atx in [26,Theorem 9.13], which says that f is locally Lipschitz continuous atx if and only if ∂ ∞ f (x) = {0} or equivalently ∂ f (x) is locally bounded atx, and in that case, In parallel fashion we can characterize the local Lipschitzian continuity of f atx relative to X by utilizing the following notion of projectional (horizon) subgradients, whose construction are motivated by applying Theorem 2.3 to the profile mapping E f . Definition 3.1 (projectional subgradients). Consider a function f : Ê n → Ê, a pointx with f (x) finite, and a convex set X withx ∈ X. For a vector v ∈ Ê n , we say that Here, x k f +δ X − −−−−−−− →x amounts to x k →x with f (x k ) → f (x) and x k ∈ X for all k. In the case that x X, we define ∂ X f (x) := ∅ and ∂ ∞ X f (x) := ∅.
By using the notion of projectional (horizon) subgradients, we can extend [26,Theorem 9.13] to deal with the Lipschitzian continuity of a function relative to some closed and convex set, and also the Lipschitz modulus. Theorem 3.1. Consider a function f : Ê n → Ê, a pointx ∈ Ê n where f is finite and locally lsc, and a closed and convex set X ⊂ dom f such thatx ∈ X. Then the following conditions are equivalent: (a) f is locally Lipschitz continuous atx relative to X.
Moreover, when these conditions hold, the following properties hold: holds for all x close enough tox in X.
(ii) The projectional coderivative of E f atx for f (x) with respect to X is given by Proof. As noted at the very beginning of the section, all the results can be verified by applying Theorem 2.3 to the profile mapping E f . Let Q := gph(E f | X ). Clearly, Q = epi( f + δ X ) and f + δ X is locally lsc atx. So by [26,Theorem 8.9], we have for all x ∈ X, and hence by definition, By Lemma 3.1, we have for all x ∈ X and α > f (x), [(a) =⇒ (b)]: By Theorem 2.3, there is some κ ≥ 0 such that the inequality holds for all (x, α) close enough to (x, f (x)) in Q (i.e., x ∈ X with f (x) ≤ α). In view of the continuity of f atx relative to X (due to the Lipschitzian continuity of f atx relative to X), the above inequality holds in particular for all x close enough tox in X with α = f (x). In combining this with the formula (38), we assert that (b) holds as the following inequality holds for all x close enough tox in X: Note that the convention proj T X (x) ∂( f + δ X )(x) := ∅ is used in (b) for the case that x X.
[(b) =⇒ (a) and (i)]: According to (b), there are some δ > 0 and τ > 0 such that Let v ∈ ∂ ∞ ( f + δ X )(x). By the definition of the horizon subgradient, there are sequences λ k ↓ 0 In view of (40), the following inequality holds for all k sufficiently large: or equivalently (as in the proof of Theorem 2.3), Let w ∈ T X (x) ∩ S. As X is closed and convex (implying that w is a regular tangent vector to X at x), it follows from regular tangent cone properties [26,Theorem 6.26] that there exists some w k ∈ T X (x k ) such that w k → w. In view of (42), we have for all k sufficiently large, implying that v, w ≤ 0 and hence v ∈ T X (x) * = N X (x). That is, (41) follows.
Next we suppose by contradiction that (a) is not fulfilled. Then by Theorem 2.3 again, there exist some ( (the inclusion due to (39)) such that the following inequality holds for all k: Clearly, we have lim sup and hence f (x k ) → f (x) (due to f being locally lsc atx). In view of (41), we have for all k sufficiently large, Thus for sufficiently large k, we deduce from (38) that λ k > 0 and hence v k λ k ∈ ∂( f + δ X )(x k ), for otherwise λ k = 0 would imply v k ∈ ∂ ∞ ( f + δ X )(x k ) and hence proj T X (x k ) (v k ) = 0, contradicting to (43). In combining this with the inequality (43), we assert that lim sup contradicting to (b). This contradiction indicates that (a) must be fulfilled. As the relative continuity of f is implied by (a), we get (i) immediately from (41).
[(b) ⇐⇒ (d)]: The equivalence follows readily from the definition of the horizon projectional subgradients in Definition 3.1.
When the properties described in (38), (a), (b) and (i) are taken into account, the formula for D * X E f (x | f (x)) in (ii) can be obtained in a straightforward way from Definitions 2.2 and 3.1. In view of the fact that f + δ X is finite and locally lsc atx, we get from [26, corollary 8.10] that there exists some x k The boundedness of ∂ X f (x) follows readily from the local boundedness in (b). That is, ∂ X f (x) is nonempty and compact. By the definition of the outer norm, we have The formula for lip X f (x) in (iii) then follows from Theorem 2.4.
[(a) =⇒ (c)]: By (a), we have lip X f (x) < +∞. Then for all x close enough tox in X, f is locally Lipschitz continuous at x relative to X with That is, (c) follows. This completes the proof.
In what follows, we consider a proper, lsc, sublinear function h on Ê n . It is well-known that there is a unique closed, convex set D in Ê n such that h = σ D , i.e., h can be expressed as the support function of D. If D is bounded, then h is finite everywhere, entailing that h is locally Lipschitz continuous everywhere, and in particular, However, when D is unbounded, h is not finite everywhere anymore, and in this case, it is more desirable to study the relative Lipschitzian property of h on dom h, which, due to the positive homogeneity of h, amounts to the local Lipschitz continuity of h at 0 relative to dom h.
In the following corollary, we apply Theorem 3.1 to fully characterize the local Lipschitz continuity of h at 0 relative to dom h by describing ∂ dom h h(0) and ∂ ∞ dom h h(0) in terms of all the exposed faces of D along with corresponding exposed faces of the horizon cone D ∞ . v, x the nonempty face of D ∞ exposed also by x. Then the equality holds for all x ∈ (D ∞ ) * with F D,x ∅. Moreover, in terms of the faces pairing F D,x and F D ∞ ,x , the following properties hold:

(a) ∂ dom h h(0) is nonempty and closed with
In particular, ∂ dom h h(0) contains the projections of D on (D ∞ ) * , as well as all the bounded exposed faces of D.

So we have
and hence T dom h (x) = (F D ∞ ,x ) * . Note that F D,x ′ = ∅ whenever x ′ ∈ (D ∞ ) * \ dom h, and that the convention proj M ∅ := ∅ is used for any nonempty set M. The closedness of ∂ dom h h(0) and the formulas in (a) follow readily from the definition of the projectional subgradients (Definition 3.1) and the positive homogeneity of h. As we have 0 ∈ (D ∞ ) * , F D, 0 = D and F D ∞ , 0 = D ∞ , ∂ dom h h(0) clearly contains the projections of D on (D ∞ ) * , and thus is nonempty. Assume that F is a bounded exposed face of D. Then by definition there is some The equivalence in (b) follows directly from Theorem 3.1 if the closedness of dom h is guaranteed on both sides. To see this, let x k → x * 0 with x k ∈ ri(dom h)\{0} for all k and we will show x * ∈ dom h for both sides.
First assume that h is locally Lipschitz continuous at 0 relative to dom h. Then we have lim sup which implies by the lower semicontinuity of h that and hence that h(x * ) < +∞ or equivalently x * ∈ dom h.
Let v k ∈ ∂h(x k ) or equivalently v k ∈ D with x k ∈ N D (v k ). Then there is someṽ k ∈ D ∩ L ⊥ such that x k ∈ N D (ṽ k ) = N D (v k ). Thus we haveṽ k ∈ F D,x k andṽ k = proj (F D ∞ ,x k ) * (ṽ k ) ∈ ∂ dom h h(0). As ∂ dom h h(0) is assumed to be bounded, {ṽ k } is clearly a bounded sequence. By taking a subsequence if necessary, we assume thatṽ k → v * . Then we have v * ∈ D with x * ∈ N D (v * ) or equivalently v * ∈ ∂h(x * ). This implies that x * ∈ dom h as expected.
The first equality in (46) follows from (b) and Theorem 3.1 in a straightforward way, while the second one follows from (a), the definition of the excess of a set over another set (see Section 1 for the definition), and the fact that the following equalities hold for all x ∈ (D ∞ ) * and v ∈ F D,x : It remains to show (d). Let x ∈ (D ∞ ) * be given arbitrarily. As D is assumed to be polyhedral, D ∞ and F D,x ∅ are also polyhedral and there exists some bounded set B x such that F D,x = B x + (F D,x ) ∞ = B x + F D ∞ ,x (the second equality due to (45)). So we have This suggests by (46) that ∂ dom h h(0) is bounded, as every polyhedral set has only finitely many faces. This completes the proof.
To end this subsection, we illustrate Corollary 3.1 by two simple examples.
Clearly, D ∞ = Ê 2 + , (D ∞ ) * = −Ê 2 + and D does not have one-dimensional face. Moreover, the projection of D on (D ∞ ) * is (0, 0) T , and each (x 1 , 1/x 1 ) with x 1 > 0 is a zero-dimensional face of D. It is clear to see from Corollary 3.1 (a) that Then by Corollary 3.1 (b), h is not locally Lipschitz continuous at 0 relative to dom h due to the unboundedness of ∂ dom h h(0). Explicitly, we have and in terms of x k := (0, −1/k) T and x ′ k := (−1/k 2 , −1/k) T , In

Level-set mappings and structural subgradients
Given a function f : Ê n → Ê, a pointx ∈ Ê n where f is finite and locally lsc, and a vector v ∈ Ê n , the level-set mapping fails to have the Lipschitz-like property at f (x) forx if and only ifv ∈ ∂ f (x). This reinterpretation of subgradients has been pointed out in [26,Theorem 9.41] by applying the Mordukhovich criterion via the coderivative Due to f being finite and locally lsc atx,f is also finite and locally lsc atx. Moreover, we havef (x) ≥ f (x) for all x, and ∂f (x) = ∂ f (x) −v for all x with f (x) < f (x) − v, x −x < +∞. Let E := gph S | X . Then, E = {(α, x) | (x, α) ∈ epif }. By [26,Theorem 8.9], we have for all (f (x), x) close enough to ( f (x),x).
We now consider the case that λ < 0. In this case, by taking a subsequence if necessary, we have proj T X (α k ) (λ * k ) < 0 for all k. This entails that α k > f (x), T X (α k ) = Ê and λ * k = proj T X (α k ) (λ * k ) < 0 for all k. Let 0 < ε k < min{α k − f (x), −λ * k } for all k. Clearly, ε k ↓ 0. In view of (−v * k , λ * k ) ∈ N epif (x k , α k ) for all k, we can find some has the Lipschitz-like property relative to .
Remark 4.1. The computation of the outer limiting subdifferential is not easy in general. However, to check the conditionv ∂ > v f (x) may be a much easier job. We note that the authors in [1] gave a concrete example in electrical circuits where the condition 0 ∂ > 0 f (x) could be checkable.
To end this section, we demonstrate by an example that the structural subgradients in ∂ > v f (x) are often located on the boundary of ∂ f (x), and that the graphical modulus lip X S ( f (x) |x) can be strictly smaller than lipS ( f (x) |x) even when both are finite.

Conclusions
By virtue of a newly-introduced projectional coderivative, we obtained a generalized Mordukhovich criterion for characterizing the Lipschitz-like property of a set-valued mapping relative to a closed and convex set, where the candidate parameter under consideration can be at the boundary of the set. We then applied this criterion to show for an extended real-valued function that its relative Lipschitzian continuity is equivalent to the local boundness of its projectional subdifferential and that for a given vector the Lipschitz-like property of the level-set mapping relative to a closed half line is equivalent to this vector being not in the outer limiting subdifferential. It is worth noting that the projection of the normal cone of the graph of the set-valued mapping onto the tangent cone of the set has played a very important role in our approach.