On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions

The Mordukhovich subdifferential, being highly important in variational and non-smooth analysis and optimization, often happens to be hard to calculate. We propose a method for computing the Mordukhovich sub-differential of differences of sublinear (DS) functions via the directed subdifferential of differences of convex (DC) functions. We restrict ourselves to the two-dimensional case mainly for simplicity of the proofs and for the visualizations. Theequivalence of the Mordukhovich symmetric subdifferential (the union of the corresponding subdifferential and superdifferential) to the Rubinov subdifferential (the visualization of the directed subdifferential) is established for DS functions in two dimensions. The Mordukhovich subdifferential and superdifferential are identiﬁed as parts of the Rubinov subdifferential. In addition, it is possible to construct the directed subdifferential in a way similar to the Mordukhovich one by considering outer limits of Fr´echet subdifferentials. The results are extended to the case of DC functions. Examples illustrating the obtained results are presented.


Introduction
The Mordukhovich subdifferential is a highly important notion in variational analysis, closely related to optimality conditions, metric regularity, Lipschitzness and other fundamental concepts of modern optimization theory (see [23,24]).This subdifferential is a closed subset of the Clarke subdifferential (see e.g.[25,Theorem 9.2]), and may be non-convex for non-convex functions, thus achieving sharper optimality conditions.In contrast to the Fréchet subdifferential (cf.[18,Example 1.1]), the Mordukhovich subdifferential of a locally Lipschitz function is always nonempty (see e.g.[22, (2.17)]).
Along with these essential advantages, there comes a substantial drawback: the Mordukhovich subdifferential is difficult to calculate even for relatively simple examples, as such computation normally involves finding the Painlevé-Kuratowski outer limit (see Section 2).For most known subdifferentials, the sum rule only has the form of an inclusion -the subdifferential of a sum is a subset of the sum of the subdifferentials [23,Theorem 3.36].This rule applied in calculations only provides a superset of the subdifferential of the sum.
We propose a method for computing the Mordukhovich subdifferential of differences of sublinear (DS) functions, which are positively homogeneous DC (difference of convex) functions, applying directed sets [2] and the directed subdifferential of DC functions [4].The DC functions represent a large family of functions.They are dense in the space of continuous functions [16] and constitute an important subclass of the quasidifferentiable functions [10].Various aspects of calculus and optimality conditions for this class of functions are discussed e.g. in [1,8,10,11,12,14,20].
The class of positively homogeneous DC functions is important enough since it contains differences of support functions and directional derivatives of DC functions.Many interesting examples of non-convex DC functions in the literature are in this class (see e.g.[4]).All results in Section 3 obtained first for DS functions can be formulated as a corollary for the directional derivative of DC functions.
The main advantage of directed subdifferentials based on directed sets is the sum rule: the directed subdifferential of a sum is equal to the sum of the directed subdifferentials [4,Proposition 4.2].This rule applied for directed subdifferentials provides the exact result.
We restrict ourselves to the two-dimensional case mainly for simplicity of the proofs and for the visualizations.Furthermore, the visualization of the directed subdifferential is essentially more complicated in dimensions higher than two, since lower dimensional mixed-type parts missing in the two-dimensional case would emerge in higher dimensions.
In this paper, the equivalence of the Mordukhovich symmetric subdifferential, the union of the corresponding subdifferential and superdifferential, to the Rubinov subdifferential (the visualization of the directed subdifferential), is established in Theorem 3.14 for the special class of DS functions in two dimensions.
While the Mordukhovich subdifferential is based on the corresponding normal cone and can be calculated by outer limits of the Fréchet subdifferential, the directed subdifferential for DC functions is essentially based on the subtraction of convex subdifferentials embedded in the Banach space of directed sets.Although these two concepts differ substantially, there are many interesting links between them.
In Theorem 3.13 we prove that certain parts of the Rubinov subdifferential comprise the Mordukhovich subdifferential.The remaining parts coincide with the Mordukhovich superdifferential (see Theorem 3.14).Furthermore, Theorem 3.11 links outer limits of the Fréchet subdifferential to the directed subdifferential.The assumption on positive homogeneity of the DC functions is dropped in Theorems 3.16 and 3.17 yielding the connection of the Rubinov subdifferential to the Mordukhovich symmetric subdifferential of the directional derivative for the broader class of DC functions.
The paper is organized as follows.In the next section we recall necessary definitions, notation and results on Fréchet subdifferential.In Sect. 3 the relation between the Mordukhovich and the directed subdifferential is discussed.We illustrate our results with several examples in Sect. 4. In the last section we sketch directions for future research.

Preliminaries
Recall that f : IR n → IR is called positively homogeneous, if f (λ x) = λ f (x) for all x ∈ IR n and λ > 0. Clearly, f (0) = 0 for positively homogeneous functions.A function is sublinear if it is convex and positively homogeneous.Recall that support functions of compact sets are sublinear.We denote by S n−1 the unit sphere in IR n , and by cl(A), co(A) the closure and the convex hull of the set A respectively.The following operations on sets A, B ⊂ IR n are well-known: The last operation is used in the definition of the Mordukhovich superdifferential and in the negative part of the visualization of the directed subdifferential.For the sets A, B ⊂ IR n the operation is called the geometric difference of the sets A and B. This difference is introduced by Hadwiger in [13] as well as in [28] and is also called Minkowski-Pontryagin difference.
Let C ⊂ IR n be nonempty, convex, compact and l ∈ IR n .Then, the support function and respectively the supporting face of C in direction l are defined by We denote by Lim sup the Painlevé-Kuratowski outer limit and by Lim inf the inner limit of sets (see [30,Chap. 4]).Intuitively, the outer limit of a sequence of sets consists of the limiting points of all converging subsequences of points from these sets.In contrast, the inner limit consists of limiting points of all sequences constructed from points taken from almost every set in a way that only a finite number of sets can be missed out.For a more rigorous definition (see [30,Sect. 4.A]), first consider the set N ♯ ∞ of all infinite subsequences in the set of natural numbers N ♯ ∞ := {N ⊂ IN | N infinite}, and the set N ∞ of all the sequences of natural numbers which include all numbers beyond a certain value, i.e.
For a set-valued mapping F : IR n → IR m and x ∈ IR n , the outer and inner limit of F as x → x is naturally defined as Lim sup Lim inf Clearly, the inner limit is a subset of the outer limit.If they are equal, this set is called the Painlevé-Kuratowski limit and is denoted by Lim k→∞ C k , respectively Lim x→ x F(x).
Remark 2.1.Let F(•) be a uniformly bounded mapping defined in a neighborhood of the point x ∈ IR n with nonempty images in a finite-dimensional space.It is easy to show that if the Painlevé-Kuratowski outer limit is a singleton Lim sup x→ x F(x) = { ȳ}, it is equal to the Painlevé-Kuratowski limit.Indeed, by the assumption, for any sequence x n → x, there is a converging subsequence y n k ∈ F(x n k ) and any such subsequence may have only the point ȳ as the limit.
The classical Moreau-Rockafellar subdifferential of a convex function f : It is well-known (see e.g.[15, Chap.V, Definition 1. where f ′ (x; l) is the directional derivative of f at x in direction l.
In the sequel, the Moreau-Rockafellar subdifferential of a sublinear function g at zero is denoted by ∂ g instead of ∂ g(0).
Also, for the unique supporting point of a supporting face we denote The Dini subdifferential (see [5,17,26,27]) of a directionally differentiable function f : The Fréchet subdifferential and the superdifferential/upper subdifferential (see [5,6,18,23]) of a function f : IR n → IR at a point x ∈ IR n are defined as follows: The Fréchet subdifferential coincides with the Fréchet gradient for a Fréchet differentiable function, and with the subdifferential for a convex function.One can think of ∂ F f ( x) and ∂ + F f ( x) as of the set of gradients of linear functions "supporting" f from below resp.above at x.While the Fréchet subdifferential is defined for a vast class of functions, and can be used to check optimality conditions, in many cases it happens to be an empty set, which is a serious drawback for applications.
The Fréchet subdifferential possesses several useful properties summarized in the following two lemmas.
Lemma 2.2.Let f : IR n → IR be positively homogeneous and l ∈ IR n .Then and f (•) is the support function of the Fréchet subdifferential, i.e.

⊓ ⊔
The following result, which is an immediate consequence of [9, Theorem 2], is used for evaluating Fréchet and Mordukhovich subdifferentials in the examples.Lemma 2.3.Let f : IR n → IR be directionally differentiable.(i) If the directional derivative of f at x can be represented as where ϕ t are sublinear functions for every t ∈ T and T is an arbitrary index set, then where ϕ t are sublinear functions for every t ∈ T , then The next lemma states that the Fréchet subdifferential coincides with the Dini one for DC functions.
Lemma 2.4.If f = g − h is DC with convex functions g and h, then Proof.Since each convex function g, h : IR n → IR is locally Lipschitz (see [15, Chap.IV, Theorem 3.1.2]),each DC function f = g − h is also locally Lipschitz.Hence, we can apply Proposition 1.16 from [18], which yields where we use the notation in [18].In our setting d f (x)(l) corresponds to the lower Hadamard directional derivative of f at x in the direction l.
Since each convex function (and hence, each DC function) is directionally differentiable, the limit inferior is indeed a limit with d f (x)(l) = f ′ (x; l).As we are dealing with finite-dimensional spaces, d w f (x; l) = d f (x; l) holds, and [18, Proposition 1.17] yields

⊓ ⊔
Clearly, for convex functions it follows that 3 The Mordukhovich and the Directed Subdifferential in IR 2 For a continuous function f : IR n → IR, the Mordukhovich (lower) subdifferential and superdifferential (upper subdifferential) can be defined as a corresponding outer limit of Fréchet subdifferentials ([23, Theorem 1.89]): The Mordukhovich symmetric subdifferential is defined as Here, the limits are in the Painlevé-Kuratowski sense.Furthermore, the connection between the Fréchet/Mordukhovich superdifferential to the corresponding subdifferential is given by the following formulas which involve the negative function and the pointwise inverse of sets, see [18,  Directed sets, offering a visualization of differences of two compact convex sets, are introduced and studied in [2,3].Here, we only sketch the main ideas and notations on directed sets in IR 2 .
The directed sets, as well as the embedding J n of convex compact sets in IR n into the Banach space of directed sets, are defined recursively in the space of dimension n.In one dimension, the directed embedded intervals are defined by the values of the support function in the two unit directions ±1, [2] as its values (the directed "supporting face"), and a continuous function a 2 (•) : IR 2 → IR (the directed "support function").This pair is parametrized by the unit vectors l ∈ IR 2 : A convex compact set A ⊂ IR 2 is embedded into the the space of two-dimensional directed sets via the embedding map J 2 composed from the natural projection π 1,2 from IR × {0} ⊂ IR 2 onto IR, and the rotation R 2l which for any unit vector l ∈ IR 2 maps the pair (l ′ , l) (with l ′ orthonormal to l) to the standard basis (e 1 , e 2 ) in IR 2 : For a directed set The last part is formed by reprojections Q 2,l of one-dimensional visualizations from IR onto the supporting lines x, l = a 2 (l) for any unit vector l ∈ IR 2 .Equipped with a norm and operations acting separately on the components of the directed sets, the space of directed sets is a Banach space.The subtraction in this space is inverse to the Minkowski addition for embedded convex compact sets.
The directed subdifferential for DC functions and its visualization, the Rubinov subdifferential, are introduced in [4] for a DC function f = g − h as i.e. it is the difference of the two embedded subdifferentials.An explicit formula for the Mordukhovich subdifferential of a positively homogeneous function as a union of Fréchet subdifferentials is obtained in the next statement.
Proof.Denote by D the right-hand side of (20).We first show that holds by (13).Further, for any l ∈ S 1 and λ > 0 we have

It remains to show that for any l, l
Again, by Lemma 2.2 for any t > 0 Without loss of generality, either x n = 0 for all n, or x n = 0 for all n.In the former case, we have v n ∈ ∂ F f (0), and by the closedness of In the latter case, without loss of generality suppose that l n = x n x n → l ∈ S 1 .Observe that by Lemma 2.2 There are two possibilities again.Without loss of generality, either l n = l for all n, or l n − l n , l • l = 0 and l, l n = 0 for all n.In the first case, by ( 21) In the second case, let . Observe that l ′ n ⊥ l, and l ′ n = 1.Since in IR 2 there are only two unit vectors perpendicular to l, we can assume l ′ n = l ′ ∈ S 1 for all n, where l ′ is one of such two vectors.We have by (21) and Lemma 2.

⊓ ⊔
The following result about the Fréchet subdifferential of a DC function follows from (11) and [14,Sect. 4] resp.[10, Chap.III, Proposition 4.1].The following lemma will be used to explicitly calculate the first term appearing in the right-hand side of (20) in Proposition 3.1.
where ∂ g(x) and ∂ h(x) are the subdifferentials of g and h respectively.
To obtain a formula for the second term in the right-hand side of (20) for sublinear functions, we show now that the subdifferential of a sublinear function in a point l = 0 is a lower dimensional supporting face.

Lemma 3.3. Let h : IR
Proof.The equality ( 24) is trivial for l = 0.It follows from [15, Chap.VI, Proposition 2.1.5]that for l = 0 and every convex function Setting x = 0 the equality follows immediately, since (7)

holds for the positively homogeneous function h(•). ⊓ ⊔
In the next two lemmas we study the last term in the right-hand side of (20) for DS functions.
Lemma 3.4.Let f = g − h, where g, h : IR 2 → IR are sublinear.Then for every l, l ′ ∈ S 1 with l ⊥ l ′ , Proof.The function f is locally Lipschitz as a difference of sublinear functions.Hence, f is Fréchet differentiable almost everywhere, and there exists a sequence {x n } n ⊂ IR 2 such that x n , l ′ > 0 for all n, x n → 0 and f is Fréchet differentiable at l + x n .The Fréchet subdifferential of f at l + x n is nonempty and coincides with the Fréchet derivative (see [18,Proposition 1.1]).Therefore, we have Observe that for sufficiently large n we have 1 + x n , l > 0 and The positive homogeneity of f together with (8) yields Observe that t n > 0 and also t n → 0, i.e. t n ↓ 0. Since f is locally Lipschitz, the sequence {∇ f (l + x n )} is bounded, hence, has a converging subsequence.This subsequence satisfies Lim sup n→∞ which yields the nonemptiness of Lim sup t↓0 ∂ F f (l + tl ′ ).

⊓ ⊔
The following result establishes that the set limit (i.e. the limit of the sequence) of the subdifferentials ∂ h(l + tl ′ ) evaluated at small orthogonal disturbances of the direction l is a singleton.This fact is needed later in the representation theorem for directed subdifferentials.Lemma 3.5.Let h : IR 2 → IR be sublinear.Then for any l, l ′ ∈ S 1 with l ⊥ l ′ , the set Y (l ′ ,Y (l, ∂ h)) is a singleton, and Lim Proof.First, we will prove the claimed equality for the outer limit Lim sup t↓0 ∂ h(l + tl ′ ), and then apply Remark By the definition of supporting face and by (7) we have and Taking limits as n → ∞ (t n ↓ 0) on both sides of ( 26) and ( 27), we obtain Subtracting ( 30) from ( 29), we have v n , l ′ ≥ ṽ, l ′ .Thus for any cluster point v Since Y (•, ∂ h) is upper semicontinuous and has closed values, it follows from (24) and 31) and the inclusion "⊂" in ( 25) is proved with the outer limit in the left-hand side.Assume now that Y (l ′ ,Y (l, ∂ h)) contains two different points ṽ1 , ṽ2 .Clearly, which contradicts the assumption that the points are different.Hence, the right-hand side in ( 25) is just a singleton and the equality follows by the non-emptiness of the left-hand side guaranteed by Lemma 3.4, Equ. ( 12) and Remark 2.1.

⊓ ⊔
Thus, (25) in the above lemma can be reformulated with the notation (5) as The previous lemma will be generalized to DC functions.The following lemma states an explicit formula for the third term appearing in the right-hand side of (20) in Proposition 3.1.Lemma 3.6.Let f = g − h, where g, h : IR 2 → IR are sublinear.Then for every l, l ′ ∈ S 1 , l ′ ⊥ l the outer limit Lim sup t↓0 ∂ F f (l + tl ′ ) is a singleton, and Then there exist sequences Therefore, for all n ∈ IN there are Since {v n } and {w n } are bounded (as they belong to the corresponding upper semicontinuous subdifferentials of g and h), the sets Lim sup n→∞ {v n } and Lim sup n→∞ {w n } of cluster points of the corresponding sequences are nonempty.Moreover, by Lemma 3.5 we have Lim sup where we have used the notation (5).Hence the sequences {v n } and {w n } converge and have unique cluster points.Therefore Since u is arbitrary, we have Lim sup holds and we obtain (33) from (34).

⊓ ⊔
For the convenience of the reader, we include a full proof for the explicit formula of the subdifferential of a sublinear function with the help of two collinear directions orthogonal to the supporting face in Lemma 3.3, although this geometric fact is rather obvious.Lemma 3.7.Let h : IR 2 → IR be a sublinear function.Then for every l, l ′ ∈ S 1 with l ′ ⊥ l, where we used again the notation (5).
Obviously, co{d h (l; −l ′ ), d h (l; l ′ )} ⊂ Y (l, ∂ h), and we only need to show the opposite inclusion.Assume that there Then by the separation theorem there exists l ∈ IR 2 such that Since the representation v = v, l • l + v, l ′ • l ′ holds, we can use v ∈ Y (l, ∂ h) as well as ( 4) and (7) to observe that Using d h (l; l ′ ) ∈ Y (l, ∂ h) twice, the equality h(l) = d h (l, l ′ ), l follows, if l, l ′ ≥ 0, as well as Analogously, if l, l ′ < 0, the following estimate is valid due to Clearly, (36) resp.(37) contradict (35), hence our assumption is wrong.

⊓ ⊔
The next two lemmas will be used in the further theorems.The first one connects the first component of the embedding (17) of convex sets into the space of directed sets to the interval which coincides with the projection of the line segment from Lemma 3.7.In the embedding the natural projection π 1,2 and the rotation R 2,l in [2] are used.
Proof.Observe that l ⊥ l ′ , so that Lemmas 3.3 and 3.7 apply with Since h(l) = d h (l; ±l ′ ), l , the following representation holds:

⊓ ⊔
The following lemma generalizes Lemma 3.8 to DS functions.To study the result of the embedded difference of subdifferentials, the convex sets in the first component of the embedding ( 17) can be calculated with the help of the two endpoints of the interval.Lemma 3.9.Let f = g − h, where g, h : IR 2 → IR are sublinear.Consider l ∈ S 1 and the orthogonal vector l ′ = R ⊤ 2,l e 1 .Then where the notation (5) and are used.
Proof.Clearly, l ⊥ l ′ .By Lemma 3.6 the two sets

⊓ ⊔
We apply the two lemmas above to represent the directed subdifferential of a positively homogeneous DC function in IR 2 with the help of outer limits of Fréchet subdifferential.The unique supporting points calculated in Lemma 3.9 are used to determine the (one-dimensional) first component of the directed subdifferential.Lemma 3.10.f = g − h, g, h : IR 2 → IR, sublinear.Then, using the notation (5), Proof.Observe that δ * (l, ∂ g) = g ′ (0; l) = g(l) by ( 4) and Lemma 2.2 and therefore,

⊓ ⊔
As a first main result, we connect the representation of the directed subdifferential to outer limits of Fréchet subdifferentials.
Theorem 3.11.Let g, h : IR 2 → IR be sublinear functions, and let f = g − h.Then the directed subdifferential of f at zero ) l∈S 1 can be constructed via limits of Fréchet normals as follows: for every l ∈ S 1 let where D − (l) Proof.By Lemma 3.9 where we used again the notation (5).Therefore, which coincides with the directed subdifferential − → A of f by Lemma 3.10.

⊓ ⊔
The equality for the Fréchet subdifferential in the next lemma will be used to explicitly calculate the second term appearing in the right-hand side of (20) in Proposition 3.1.Geometrically, this fact is easy to believe so the reader may skip the technical proof.Lemma 3.12.Let f = g − h, where g, h : IR 2 → IR are sublinear.Then for every l ∈ S 1 where we used again the notation (5) and and case 2 is given, if the inequality ">" holds.
It holds that Let us denote for abbreviation Let us study the scalar product of u ∈ (∂ g(l) − g(l)l) − * (∂ h(l) − h(l)l) and η ∈ IR n : Both shifted line segments are spanned by the vector l ′ , hence the geometric difference lies also in this span which is demonstrated by setting η = ±l in the above inequality: l, u ≤ 0 − 0 = 0, −l, u ≤ 0 − 0 = 0 Hence, l, u = 0. Let us study the scalar product in the orthogonal directions l ′ and −l ′ .
Then, equations (42) and (43) yield the contradiction We will show that "⊂": Let η ∈ IR n .Using l, u = 0 and the orthonormal basis {l, l ′ }, we get η Hence, the first endpoint of the line segment M 2 lies in M 1 : Now, we proceed similarly with the second endpoint (µ 2 − ν 2 )l ′ .Since (44) and µ 2 − ν 2 + ν 1 ≥ µ 1 is valid, An immediate consequence is that the second endpoint of M 2 also lies in M 1 : This equality for both sets is used to reformulate the geometric difference: Let us now calculate both endpoints of the line segment using (41): This finally shows that

⊓ ⊔
The next main theorem shows that the Mordukhovich subdifferential of f at 0 can be represented via visualization parts from ( 18)-( 19) of the directed subdifferential.Theorem 3.13.Let f = g − h, where g, h : IR 2 → IR are sublinear functions, and let where bd denotes the boundary of a set in IR, and is the reprojection as in [3].Proof.First of all, observe that by Proposition 3.1 where l ′ = R ⊤ 2,l e 1 .The proof consists of three parts: Step 1: We will show that the positive part coincides with the Fréchet subdifferential at x = 0: Step 2: We will conclude that the reprojected positive part is the second term in (46): Step 3: We will prove the following equality for the reprojected boundary points: It is not difficult to see that Steps 1-3 together with (46) yield (45).
Step 1: For the Fréchet subdifferential, Lemma 2.2 yields This equation can be compared with the definition of the positive part of the directed set: 50) and (51) we conclude (47).
Step 2: By Lemma 3.12 for all l ∈ S 1 we have where the notation ( 5) is again used.At the same time, Lemma 3.10 yields for every l ∈ S 1 Observe that and hence From ( 52), ( 53) and (55) the equation (48) follows.

⊓ ⊔
The Mordukhovich superdifferential and symmetric subdifferential of f at 0 is represented via the directed subdifferential in the following theorem.Besides isolated points from the reprojected lower dimensional positive part of the directed subdifferential, the Mordukhovich superdifferential forms the negative two-and one-dimensional part in the visualization of the directed subdifferential.The positive parts are reflected by the Mordukhovich subdifferential (see Theorem 3.13) so that the Mordukhovich symmetric subdifferential form the complete visualization of the directed subdifferential for DS functions.Theorem 3.14.Let f = g − h, where g, h : IR 2 → IR are sublinear functions, and let Proof.Apply Theorem 3.13 to − f = h − g and use [3, Proposition 3.8]: Together with (15), equation (60) follows immediately.Since and ( 18) hold, the second equation (61) follows easily.

⊓ ⊔
Applying the previous Theorems 3.11, 3.13 and 3.14 to the directional derivative generalizes these results to the class of general DC functions (which are not necessarily positively homogeneous).
As a starting point, we will demonstrate that the directed subdifferential of the function at x coincides with the one of its directional derivative evaluated at direction l = 0. Proposition 3.15.Let f = g − h with g, h : IR n → IR be convex functions.Then, Proof.By Lemma 3.3, the convex subdifferential of g ′ (x; ) in 0 coincides with the one of g(x): The same is true for the convex function h such that

⊓ ⊔
Since the Mordukhovich subdifferential of the directional derivative may differ from the one for the function itself (see Example 4.3) in contrary to the directed subdifferential, the following results for the Mordukhovich subdifferentials have to be formulated with the directional derivative.The next theorem yields the connection between outer limits of Fréchet subdifferentials and the directed subdifferential.Theorem 3.16.Let g, h : IR 2 → IR be convex functions, and let f = g − h.Then the directed subdifferential − → A = ( − → A 1 (l), a 2 (l)) l∈S 1 of f at x can be constructed via limits of Fréchet normals as follows: for every l ∈ S 1 let Proof.Applying [10, Sect.I.3, Proposition 3.1], the directional derivative is a DS representation.Hence, Proposition 3.15 and Theorem 3.11 can be applied.

⊓ ⊔
The next theorem for DC functions, in which we can drop the assumption of positive homogeneity, could be seen as the non-convex counterpart of the following result for locally Lipschitz and directionally differentiable function in [19,Sect. 3] and [8, (35) where ∂ MP f (x) is the Michel-Penot subdifferential of f in x (see [8, 21]).In what follows the Mordukhovich symmetric subdifferential for the directional derivative at x in direction 0 coincides with the Rubinov subdifferential at x, i.e. its visualized directed subdifferential.
Theorem 3.17.Let f = g − h, where g, h : IR 2 → IR are convex functions, and let Proof.As in the proof of Theorem 3.16 the equality (62) of the directed subdifferential of f ′ (x; •) in 0 and the one of f (•) in x holds.The claimed equalities are proved by applying Theorems 3.13 and 3.14.⊓ ⊔ Remark 3.18.All the lemmas starting from Lemma 3.3 could be adapted to the convex (instead of sublinear) situation.For this purpose, the function must be replaced by its directional derivative, which is sublinear with respect to its second argument.E.g., Lemma 3.5 reads for h being only convex:

Examples
For each of the presented examples we will first calculate theoretically the Mordukhovich subdifferential and superdifferential.Their union, the symmetric subdifferential is compared visually with the Rubinov subdifferential in [4].We will frequently use Lemma 2.3 for evaluating the Fréchet subdifferential which is a basic tool for calculating the Mordukhovich subdifferential with (13).Analogously, we proceed with the Fréchet superdifferential and ( 14) in the same way to evaluate the Mordukhovich superdifferential.
The first example is governed by a parameter r by which three different cases could be studied: the Mordukhovich subdifferential has nonempty interior (r = 0.5), the Mordukhovich superdifferential has nonempty interior (r = 2.0) and both have empty interior (r = 1.25).This corresponds to nonemptiness of the positive part resp. of the negative part as well as the mere presence of the mixed-type part in the directed subdifferential.
To evaluate the Mordukhovich lower/upper/symmetric subdifferential of f at zero directly, we first need to calculate the Fréchet subdifferentials of f at zero and in its neighborhood.
B. The Fréchet superdifferential at 0. Observe that where where B r (m) = {x | x − m = r}.Using (10), the Fréchet superdifferential can be calculated as C. The Fréchet sub-and superdifferentials around 0. For every x = 0 the function h is smooth, hence For the Fréchet superdifferential in all points x = 0 we have otherwise, since g is not Fréchet superdifferentiable due to [18,Proposition 1.3].
Observe that for x = 0, the subdifferential of g is given by and It is not difficult to observe that for every l ∈ S 1 with l 1 l 2 = 0, we have by applying ( 73) and (74).For l = (1, 0) and l ′ = (0, 1) The corresponding outer limits for the remaining directions can be evaluated analogously.We have From (68), (73), ( 75) and (76) the Mordukhovich subdifferential is given by Analogously, from (72) and ( 74)-( 76) The Mordukhovich subdifferentials of f at 0 for the values of r = 0.5, 1.25 and 2.0 are plotted in Figs.1-3.
The corresponding series for the visualization of the directed subdifferentials with the same values of r are plotted in Fig. 4, see also [4,Example 5.7] for further explanations.The plots coincide with the pictures of the Mordukhovich symmetric subdifferentials.Since the subdifferentials of the convex functions g and h are known, the Rubinov subdifferential could be easily calculated as the visualization of the difference of these embedded convex sets.
The arrows in Fig. 4 indicate outer normals to the directed "supporting faces".They also form the parametrizing directions in (16) for the directed subdifferential.The positive part in the left picture of Fig. 4 is a convex set.It is coloured in gray and only outer normals are attached to its boundary.The other non-convex part belongs to the mixed-type part.Similarly for the right picture in Fig. 4. The gray convex subset is the negative part and has only inner normals attached to its boundary.The positive and negative part in the middle picture are empty and the Rubinov subdifferential consists only of the mixed-type part.Note that the unique "supporting points" belong both to the Mordukhovich subdifferential and superdifferential due to Theorems 3.13 and 3.14, since for such a point the lower dimensional positive and negative parts coincide with the point itself.
As in Example 4.1, one can see that the four unique directed "supporting points" (±1, ±1) (see Fig. 5) are present both in the Mordukhovich subdifferential and superdifferential.The only segment co{(−1, −1), (1, −1)} in the Mordukhovich superdifferential may be recognized from the Rubinov subdifferential in Fig. 6 as coming from a negative part of a directed interval, since there are outer normals attached to its ends (see Fig. 6) where the projections are pointing inside the interval, contrary to all the segments in the Mordukhovich subdifferential.
Also the Rubinov subdifferential (the visualization of the directed one in Fig. 6) coincides with the Mordukhovich symmetric subdifferential, according to Theorem 3.14.The last example shows the difference between Theorems 3.13/3.14and 3.17.In this example the function f is DC, but not positive homogeneous.So we cannot expect that we have equality between the Mordukhovich symmetric subdifferential and the Rubinov one (the visualization of the directed subdifferential) as in Theorem 3.14.
A similar reasoning shows that the Fréchet subdifferential and superdifferential of the directional derivative coincides with the gradient of f ′ (0; •) with respect to l in any direction η by [18,Proposition 1.3].Hence, the Mordukhovich subdifferential and the Mordukhovich superdifferential also equal to the point (0, 1) due to (14).

Conclusions
As we have shown in this paper, the connection between the Mordukhovich subdifferential/superdifferential and the Rubinov subdifferential may provide substantial information related to their computing and in their applications.This relation will be investigated and explored in more details in our further research.Especially, we are currently working on the extension of our results from the class of DC functions to quasidifferentiable functions and on their application to quasidifferential calculus.Another focus of future research will be the case of dimension higher than two.
c∈C l, c .Note that for l = 0, Y (l,C) = C.By y(l,C) we denote any point of the set Y (l,C), and if the latter is a singleton (i.e., there is a unique supporting point), then Y (l,C) = {y(l,C)}.The supporting face Y (l,C) equals the subdifferential of the support function of C at l [29, Corollary 23.5.3].