Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces

For a set-valued mapping M defined between two Hausdorff topological vector spaces E and F and with closed convex graph and for a given point (x, y) ∈ E × F, we study the minimal time function associated with the images of M and a bounded set Ω ⊂ F defined by T M,Ω (x, y) := inf{t ≥ 0 : M(x) ∩ (y + tΩ) ̸ = 0}. We prove and extend various properties on directional derivatives and subdifferentials ofT M,Ω at those points of (x, y) ∈ E×F (both cases: points in the graph gphM and points outside the graph). These results are used to prove, in terms of the minimal time function, various new characterizations of the convex tangent cone and the convex normal cone to the graph ofM at points inside gphM and to the graph of the enlargement set-valued mapping at points outside gphM. Our results extend many existing results, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces (Bounkhel, 2012; Bounkhel andThibault, 2002; Burke et al., 1992; He and Ng, 2006; and Jiang and He 2009). An application of the minimal time functionT M,Ω to the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper.

From the above cases, we can see the importance of the study of the minimal time function T ,Ω in normed vector spaces as well as in Hausdorff topological vector spaces.This type of study will unify the study of all the above functions.
The case of minimal time function associated with a closed set  has been the subject of many recent works [1-3, 5-10, 12, 13].To the best of our knowledge, the unique work studying the function T ,Ω is [16], in which the author studied the Fréchet subdifferential of T ,Ω in Banach spaces.We mention that there are no results on the directional derivatives and subdifferentials of T ,Ω in the Hausdorff topological vector spaces.Starting from this point, as a goal, we will develop a thorough study of the minimal time function in Hausdorff topological vector spaces in the convex setting.The nonconvex case will be the subject of a series of forthcoming works by the author.In the present paper, we extend various existing results on directional derivatives and subdifferentials of T ,Ω and their relationships to tangent and normal cones in Hausdorff topological vector spaces.The paper is organized as follows.Section 2 is devoted to state the main notations and definitions used throughout the paper.In Section 3, we prove our main results for points on the graph of .The case of points outside the graph of  is studied in Section 4. In the last section we state an application of the minimal time function T ,Ω to the study of the calmness property of optimization problems in Hausdorff topological vector spaces.

Notations and Preliminaries
Throughout the whole paper (unless otherwise specified), we assume that  and  are two Hausdorff topological vector spaces.We will denote by  * and  * the topological dual of  and , respectively, and by ⟨(⋅, ⋅), (⋅, ⋅)⟩ the pairing between the spaces  ×  and  * ×  * .

Points on the Graph of the Set-Valued Mapping
Before starting the study of minimal time functions for setvalued mappings with closed convex graphs, we need to prove some results for general set-valued mappings  :    with nonempty values (with graph not necessarily closed nor convex).These results have their own interests.We start with the following lemma which is needed in all the proofs of our work.

Lemma 1.
Let  and  be Hausdorff topological vector spaces.Assume that Ω is a bounded set in  and  is a set-valued mapping with nonempty values in .
Note that the assumption 0 ∈ int Ω in (3) cannot be removed when the values of  are open sets.Take () as any open set, and take Ω = {0}.Clearly, for all  ∈ bd(()) As a corollary of parts ( 1) and ( 2) in Lemma 1 we have the following.

Corollary 2.
Let  and  be Hausdorff topological vector spaces, and let  :    be a set-valued mapping with nonempty closed values.Assume that Ω is a bounded set in .Then T ,Ω (, ) = 0 if and only if  ∈ ().
The following lemma characterizes the convexity of the graph of set-valued mappings in terms of the convexity of its associated minimal time function in Hausdorff topological vector spaces.It extends the well known characterization of the convexity (see, for instance, the lemma on page 53 in [14]) in terms of the distance function in normed vector spaces as well as the one in terms of the indicator function.Lemma 3. Let  and  be two Hausdorff topological vector spaces,  :    a set-valued mapping with nonempty closed values, and Ω a bounded convex set in .Then, T ,Ω (⋅, ⋅) is convex on its domain if and only if the graph ℎ  is convex.
Proof.Assume that T ,Ω (⋅, ⋅) is convex on its domain; that is, for all ( 1 , where On the other hand, by the convexity of Ω, we have which ensures that This ensures that for any  > 0. Thus, taking  → 0 completes the proof. Note that the convexity of Ω, in the proof of Lemma 3, is needed only in one direction (reverse implication); that is, the convexity of T ,Ω ensures the convexity of the graph of  even when Ω is not convex.Now, we are looking for the lower semicontinuity of the minimal time function in Hausdorff topological vector spaces.
Proposition 4. Let  and  be two Hausdorff topological vector spaces.Assume that Ω is compact in  and ℎ  is closed in  × .Then T ,Ω is lower semicontinuous at any (, ) ∈ dom T ,Ω .
By the definition of the minimal time function, we can find for any  ∈  a real number   such that Hence, for any  ∈ , there exists   ∈ Ω such that (  ,   +     ) ∈ ℎ .Using the compactness of Ω, we get the convergence of a subnet of (  )  to some point  ∈ Ω.
An inspection of the proof of the previous proposition shows that the conclusion is still valid under the assumptions that Ω is weakly compact and ℎ  is weakly closed.Consequently, we have the two following corollaries.
Corollary 5. Let  and  be two Hausdorff topological vector spaces.Assume that Ω is weakly compact in  and ℎ  is weakly closed in  × .Then, T ,Ω is lower semicontinuous at any (, ) ∈ dom T ,Ω .Corollary 6. Assume that  is a reflexive Banach space, Ω is closed convex bounded set in , and ℎ  is closed convex set in  × .Then T ,Ω is lower semicontinuous at any (, ) ∈ dom T ,Ω .
The following lemma is technical and is needed in some forthcoming proofs.Its proof is straightforward.Lemma 7. Let  and  be two Hausdorff topological vector spaces,  :    a set-valued mapping, and Ω a nonempty set in .Then, Now, we are going to establish our main results of this section.We prove some formulas and relationships between the directional derivative and the convex subdifferential of T ,Ω and the convex tangent cone and the convex normal cone of ℎ  in Hausdorff topological vector spaces at points in ℎ .We associate with the set-valued mapping  a new set-valued mapping T :    with graph ℎ T := (ℎ ; (, )); that is,  ∈ T (ℎ) if and only if (ℎ, ) ∈ (ℎ ; (, )).Theorem 8. Let  and  be two Hausdorff topological vector spaces and  :    a set-valued mapping.Assume that ℎ  is a nonempty closed convex set in  × , (, ) ∈ ℎ , and Ω is a bounded convex set in .Then, one has (1) (2) (3) if, in addition, (, ) ∈ (dom T ,Ω ),  ×  is locally convex, then (5) and Proof.
On the other hand we clearly have T T ,Ω (ℎ, ) ≤  Ω ( − ), for all  ∈ T (ℎ).Consequently, we obtain for all  that Taking the limit on this inequality, we obtain that thus completing the proof of ( 3).
( ( The last equality follows from the fact that  ×  is locally convex.Thus, the proof of the theorem is complete. Many corollaries can be deduced from this theorem.We state the following one [18] by taking Ω to be the closed unit ball of a normed vector space  and  to be a constant setvalued mapping; that is, () = .Corollary 9. Assume that  is a normed vector space,  is a closed convex subset in , and  ∈ .Then, one has Clearly, the graph of   is defined as ℎ   := {(, ) ∈  ×  : T ,Ω (, ) ≤ }.The following lemma is needed.It extends many existing results from normed spaces to Hausdorff topological vector spaces and from the case of sets to the case of set-valued mapping (see for instance [15,19]).Obviously, we have  ∈   () and () ⊂   ().We have also to point out that, due to Lemma 3, the convexity of the graph ℎ  ensures the convexity of the graph of   whenever Ω is bounded convex.However, the l.s.
The previous lemma extends Lemma 3.4 in [2] from the case of sets to the case of set-valued mappings in Hausdorff topological vector spaces.Also, it extends the inequality (4.41) on page 97 in [15] from the case where  is a normed vector space to the case where  is a Hausdorff topological vector space and from the case of distance function to images Δ  to the case of minimal time function T ,Ω .
The first consequence of Lemma 10 is the following proposition in which we establish a relationship between the directional derivatives of T ,Ω and T   ,Ω .Proposition 11.Let  and  be two Hausdorff topological vector spaces.Assume that  has closed convex graph, (, ) ∉ ℎ , and Ω is a bounded convex set.Let  := T ,Ω (, ) > 0.Then, for any (ℎ, ) ∈  × , one has Proof.It follows, directly from the first part of the previous lemma, the fact that T   ,Ω (, ) = 0 and the definition of the directional derivative.
In the following theorem, we characterize the convex tangent cone of the graph of the enlargement set-valued mapping   as the set of all directions (ℎ, ) in × for which the directional derivative of  ,Ω is nonnegative.(64) Let (ℎ, ) ∈  ×  with T  ,Ω ((, ); (ℎ, )) ≤ 0. Let (  )  be a net in (0, ∞) converging to 0 and satisfying the limit Use the lower semicontinuity of T ,Ω at (, ) and the assumption  ∉ () to find some  0 ∈  such that T ,Ω ( +   ℎ,  +   ) > 2 Therefore, and so Now, the boundedness of Ω ensures the convergence of 2    to zero, which gives the convergence of   to .Thus, by the closedness of the convex tangent cone, we get (ℎ, ) ∈ (ℎ   , (, )), and hence, the proof is finished.
The following corollary extends Corollary 1 in [14] from the case of sets to the case of set-valued mappings and from normed vector spaces to Hausdorff Topological vector spaces.It says that if we put the calmness of T ,Ω instead of its l.s.c. and the assumption 0 ∉ T ,Ω (, ) instead of the assumption T ,Ω (, ) ̸ = 0 in Theorem 14 we may remove the weak star closedness in the second equation in Theorem 14. Recall that  is said to be calm at  if there exists a closed balanced neighborhood  of zero and  ∈ N() Proof.Using the calmness of T ,Ω at (, ) and Banach-Alaoglu theorem (see Theorem 3.5 in [21]), we get the weak star compactness of   T ,Ω (, ) in  * ×  * .Therefore, the assumption 0 ∉   T ,Ω (, ) with the weak star compactness of   T ,Ω (, ) ensures the weak star closedness of the cone generated by   T ,Ω (, ); that is, cl  * (R +   T ,Ω (, )) = R +   T ,Ω (, ).Thus, the conclusion follows from Theorem 14.
The next results depend on the nonemptiness of the minimum set for the minimal time function T ,Ω defined as follows We begin with the following lemma.(89) Put  :=  Ω ( − ) > 0, and fix any  ∈ (0, 1].Clearly, by definition of the Minkowski function, we can find for any  > 0 some  0 ≥ 0 such that  −  ∈  0 Ω and  0 ∈ [,  + ), and so which ensures that Taking  → 0 + finishes the proof of the first inequality.By the first part, we have T ,Ω (,  + (1 − )) < ∞ whenever T ,Ω (, ) < ∞ and so for any  > 0, we can find some  ∈ Ω, some  > 0, and some  0 > 0 such that and so by taking  → 0 + , we obtain that This completes the proof of the lemma.
We use this lemma to prove the following proposition on directional derivatives and convex subdifferentials of T ,Ω at points outside the graph.
The following theorem establishes another relationship between the convex subdifferential of T ,Ω and the convex normal cone of   at points outside the graph ℎ .

Application of T 𝑀,Ω : Calmness and Exact Penalization
The primary goal in the present section is to make clear that the scalar function T ,Ω can also be a powerful tool in the study of the calmness property of optimization problems in Hausdorff topological vector spaces.Here, we are interested in the concept of calmness of perturbed optimization problems with a constraint defined by a set used and studied by Burke [22,23] in normed vector spaces.We will adapt his definition for a general perturbed problem with a constraint defined by a set-valued mapping in Hausdorff topological vector spaces, and we will prove that it is equivalent to the existence of an exact penalization in terms of the minimal time function associated with the set-valued mapping defining the constraint of the problem.
Consider the problem (P), which consists in minimizing the function  over all  ∈  satisfying 0 ∈ (), (P) { minimize  () subject to 0 ∈  () , where  :    is a closed set-valued mapping between two normed vector spaces  and  and  :  → R ∪ {+∞} is an extended real-valued function.We begin with the definition of calmness.
Definition 19.Let , , , and  be as in the statement of (P), and consider the following perturbed problem Let  :    be the feasible set-valued mapping associated with (P  ); that is,  () := { ∈ dom  :  ∈  ()} . (119) Let (, ) ∈ ℎ  and let Ω be a bounded set in .One will say that the problem (P  ) is calm at  with respect to Ω if there exist a constant  ≥ 0 and  ∈ N  () such that for every  ∈  and any  ∈ () one has  () ≤  () +  Ω ( − ) . (120) The constant  and  are called the modulus of calmness and neighborhood of calmness for (P  ) at , respectively.Remark 20.When  is assumed to be normed, the above definition coincides with the definition used in [22,23] by taking Ω to be the closed unit ball in .Observe that if (P  ) is calm at  with respect to a bounded set Ω, then  is necessarily a local solution to (P  ).
For any problem (P  ), any real number  ≥ 0, and any bounded set Ω, we will associate the function  ,,Ω defined by  ,,Ω () :=  () + T ,Ω (, ) . (121) In the following theorem we state our main result in this section.It establishes a relationship between the calmness property and the existence of an exact penalization of the general perturbed problem (P  ) in terms of the minimal time function to images associated with the set-valued mapping  defining the constraint of the problem.Theorem 21.Let (, ) ∈ ℎ .If (P  ) is calm at  with respect to Ω with modulus  and neighborhood  ∈ N  (), then  is a minimum over  of the function  ,,Ω , for all  ≥ .If, in addition, Ω is convex and 0 ∈ Ω, the converse holds.