Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 707603, 13 pages

http://dx.doi.org/10.1155/2013/707603

Research Article

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

Messaoud Bounkhel

Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 8 July 2013; Revised 30 August 2013; Accepted 2 September 2013

Academic Editor: Ismat Beg

Copyright © 2013 Messaoud Bounkhel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

For a set-valued mapping defined between two Hausdorff topological vector spaces and and with closed convex graph and for a given point , we study the minimal time function associated with the images of and a bounded set defined by . We prove and extend various properties on directional derivatives and subdifferentials of at those points of (both cases: points in the graph 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 of at points inside and to the graph of the enlargement set-valued mapping at points outside . Our results extend many existing results, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces (Bounkhel, 2012; Bounkhel and Thibault, 2002; Burke et al., 1992; He and Ng, 2006; and Jiang and He 2009). An application of the minimal time function to the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper.

1. Introduction

Let and be two Hausdorff topological vector spaces and a bounded nonempty subset of . Let be a set-valued mapping with nonempty closed values. We associate to with and the so-called minimal time function defined by

The case is a fixed closed subset of that coincides with the minimal time function associated with and and is defined by

This function plays an important role in variational analysis, optimization, control theory, Hamilton-Jacobi partial differential equations, approximation theory, and so forth; the reader can find more discussions in [1–13] and the references therein. The value can be seen as the minimal time associated with the differential inclusion , almost everywhere on , with and ; in other words,

We notice that the minimal time function covers many crucial functions in variational analysis: the distance function to images , the indicator function, the usual distance function, and the Minkowski function, by taking some particular cases of and as follows.

(1)If , then which coincides with the Minkowski (gauge) function associated with ; that is,

(2)If , then , which coincides with the indicator function associated with . Here, if and , otherwise.(3)If is normed, , and is the closed unit ball in , then .(4)If is normed and , then

which coincides with the distance function associated with the images of (which has been successfully used in optimization theory, first by Clarke in [14] for Lipschitz set-valued mappings and later by various authors (see for instance the book [15] and the references therein) for set-valued mappings that are not necessarily locally Lipschitz).(5)If is normed, , and , then

which coincides with the usual distance function associated with in .

From the above cases, we can see the importance of the study of the minimal time function 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 is [16], in which the author studied the Fréchet subdifferential of in Banach spaces. We mention that there are no results on the directional derivatives and subdifferentials of 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 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 to the study of the calmness property of optimization problems in Hausdorff topological vector spaces.

2. 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 .

Let be a nonempty closed convex subset of and a point in . The convex tangent cone is defined by , where denotes the set of all nonnegative real numbers. The convex normal cone to at is defined as the negative polar of ; that is, , where ,  for  all  , for any subset . Recall also from [14] that is also characterized by ,  for  all  .

Let be a convex function from into with ; the directional derivative (resp., the convex subdifferential) of at is defined by

Note that for l.s.c convex functions we have , whenever and is a Hausdorff locally convex topological vector space. Here, denotes the support function associated with a closed subset defined by and is called the algebraic interior of (for more details on the core, we refer the reader to [17]).

3. Points on the Graph of the Set-Valued Mapping

Before starting the study of minimal time functions for set-valued 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 is a bounded set in and is a set-valued mapping with nonempty values in .

(1)If , then .(2)Conversely, for any , one has .(3)If in addition , then for any one has .

Proof. (1) Let with , then for all , such that with ; that is, there exists such that . Let be any neighborhood of in . There exists as a balanced neighborhood of such that . Since is bounded, there exists such that . Hence, there exists such that . Clearly, for any , we have , and so by the fact that is balanced we obtain

Since is arbitrary, we get the convergence of the sequence to zero, that is, the convergence of the sequence to , and since , we get .

(2) Let and let . Clearly, we have . This ensures by definition that .

(3) Assume now that . Let and fix any . Clearly, , and hence, . It follows that which ensures that .

Note that the assumption in (3) cannot be removed when the values of are open sets. Take as any open set, and take . Clearly, for all , we have .

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 is a bounded set in . Then 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 a bounded convex set in . Then, is convex on its domain if and only if the graph is convex.

Proof. Assume that is convex on its domain; that is,

for all , and all . Therefore, by Corollary 2, for any , we have , and so by the previous inequality, we obtain , and hence by Corollary 2, once again we get ; that is, , which means that is convex. Conversely, let (), and let any . Fix any . Since and , we can find that , () and () such that and (). So the convexity of ensures that . Hence,

where . On the other hand, by the convexity of , we have

which ensures that

This ensures that

for any . Thus, taking completes the proof.

Note that the convexity of , in the proof of Lemma 3, is needed only in one direction (reverse implication); that is, the convexity of ensures the convexity of the graph of even when 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 is compact in and is closed in . Then is lower semicontinuous at any .

Proof. Let . We have to prove that

The case is obvious, so we assume that

Let be a net satisfying the above ; that is, and . Let be a net of positive real numbers satisfying . By the definition of the minimal time function, we can find for any a real number such that

Hence, for any , there exists such that . Using the compactness of , we get the convergence of a subnet of to some point . Without loss of generality the subnet is still denoted by . Therefore, (by the closedness of ). This shows that , which ensures that , and hence, the proof is complete.

An inspection of the proof of the previous proposition shows that the conclusion is still valid under the assumptions that is weakly compact and is weakly closed. Consequently, we have the two following corollaries.

Corollary 5. Let and be two Hausdorff topological vector spaces. Assume that is weakly compact in and is weakly closed in . Then, is lower semicontinuous at any .

Corollary 6. Assume that is a reflexive Banach space, is closed convex bounded set in , and is closed convex set in . Then is lower semicontinuous at any .

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 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 and the convex tangent cone and the convex normal cone of in Hausdorff topological vector spaces at points in . We associate with the set-valued mapping a new set-valued mapping with graph ; that is, if and only if .

Theorem 8. Let and be two Hausdorff topological vector spaces and a set-valued mapping. Assume that is a nonempty closed convex set in , , and is a bounded convex set in . Then, one has

(1)

(2)

(3)if, in addition, , is locally convex, then

(4)

(5)and

Proof. (1) By Lemma 3, we have the convexity of , and so is well defined and hence, Corollary 2 and Lemma 7 complete the proof of part (1).

(2) Let with . Then, by definition of the convex normal cone and by Corollary 2, we have

Assume now that , and let . If , then the previous inequality holds obviously. Assume that . By definition of minimal time function, there exists and such that

Hence, we get by the definition of convex normal cones that

Therefore, combining this inequality with the assumption , we obtain that

which ensures, by taking , that

Thus, this inequality holds for any which means that .

Conversely, let . Then, by definition, we have

Using Corollary 2, we have

This means that . We have to prove that . Let and . Then, ; that is, , which ensures that . Therefore, by (28), we have

which ensures that

This gives , and hence, the proof of (2) is complete.

(3) Let . Assume first that . Then, by Corollary 2, we have . Also, we have , for all . Hence, by (2), we have , for all , and so . Now we use the assumption to write and so . Hence by Part (1) we obtain the needed equality

Assume now that . Without loss of generality, we assume that . Put . Fix any with . There exists such that ; that is, . Therefore, by Part (2), for any , we have and , and hence, we get the following

This ensures that

and since is taken arbitrarily, we take the infimum to obtain that

Let us prove the reverse inequality

Obviously, we assume that . Put . Let be a net of positive real numbers converging to zero and satisfying the limit in the definition of the directional derivative; that is, . Let and such that

Since is closed convex, we have , and so , for all ; that is, , for all .

On the other hand we clearly have , for all . Consequently, we obtain for all that

Taking the limit on this inequality, we obtain that

thus completing the proof of (3).

(4) The inclusion follows from the assertion (3) and Corollary 2. Conversely, let with . Let be a net of positive real numbers converging to zero and satisfying the limit . For any , we can find that and such that

Since the graph of is closed convex, we obtain that

On the other hand, we have , and hence, by the boundedness of the set , we get the convergence of to zero. This ensures the convergence of which ensures by the closedness of the tangent cone that , and hence, the proof of (4) is finished.

(5) The inclusion follows directly from part (2). So we prove the reverse one. To do that, we start by proving the inclusion

Fix any . Then

Since (because ) and , then . Thus

Since the reverse inequality follows directly from part (1), then we get the equality form , and hence, part (4) ensures that . Therefore, . Since clearly we have , then

which ensures that

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 to be the closed unit ball of a normed vector space and to be a constant set-valued mapping; that is, .

Corollary 9. Assume that is a normed vector space, is a closed convex subset in , and . Then, one has

4. Points outside the Graph of the Set-Valued Mapping

Now, we consider those points outside the graph of set-valued mappings with nonempty closed values. Let . Clearly, by Corollary 2, we have . Denote , and define the enlargement set-valued mapping

Clearly, the graph of is defined as . 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 ensures the convexity of the graph of whenever is bounded convex. However, the l.s.c. of ensures the closedness of the graph of . It follows then, from Lemma 3, the convexity of the minimal time function whenever is lower semicontinuous and is bounded convex.

Lemma 10. Let and be two Hausdorff topological vector spaces. Assume that has nonempty closed values, , and is a bounded convex set. Then, for any we have

and for any we have

Proof. First we prove the inequality

Assume that with . The case is obvious. Fix any satisfying . Then, there exists such that ; that is, . Then, for any , we can choose and such that

By convexity of , we have , and hence,

Therefore,

Since is taken arbitrarily satisfying , we obtain by taking the infimum over all the inequality

and by taking we get the first desired inequality

Assume now that , and let us prove the reverse inequality; that is, . The case is obvious so we suppose that . Take arbitrarily a nonnegative number for which . Then, there exists such that . Clearly, , and since , we have . Therefore,

and so

Thus,

Then,

Since is taken arbitrarily with , we can take the infimum over all those , and hence, we obtain that

The proof is complete.

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 to the case of minimal time function .

The first consequence of Lemma 10 is the following proposition in which we establish a relationship between the directional derivatives of and .

Proposition 11. Let and be two Hausdorff topological vector spaces. Assume that has closed convex graph, , and is a bounded convex set. Let . Then, for any , one has

Proof. It follows, directly from the first part of the previous lemma, the fact that 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 in for which the directional derivative of is nonnegative.

Theorem 12. Let and be two Hausdorff topological vector spaces. Assume that has a closed convex graph, , and is a bounded convex set. Assume that is l.s.c. at . Then, one has

Proof. The first equality follows directly from part (4) in Theorem 8. The direct inclusion of the second equality follows directly from the previous proposition. So it remains to prove the inclusion

Let with . Let be a net in converging to and satisfying the limit

Use the lower semicontinuity of at and the assumption to find some such that

Using now the assumption ensures the existence of some such that

Thus, combining (66) and (67), we obtain

Use now the definition of the minimal time function to choose for any some and such that

Hence,

Let and . Then,

So ; that is, .

Therefore,

and so

Now, the boundedness of ensures the convergence of to zero, which gives the convergence of to . Thus, by the closedness of the convex tangent cone, we get , and hence, the proof is finished.

Various corollaries can be deduced from the above theorem by taking special cases of and and the spaces and . We state the following one which has been proved in Corollary  4.4 in [15] (see also [20]) by taking to be a normed vector space, to be the closed unit ball of , and to be a constant set-valued mapping; that is, . In this case, coincides with the enlargement set defined by .

Corollary 13. Let be a normed vector space, a closed convex subset in , and . Then, one has

Now, we turn to establish some relationships between the convex normal cone to and the convex subdifferential of at .

Theorem 14. Let and Hausdorff topological vector spaces, a set-valued mapping with nonempty closed values, a bounded convex set in , and . Assume that is l.s.c. at , and are locally convex, and . Then,

Proof. The first equality follows directly from Theorem 8. We will prove the equality . Assume that . First, we prove that

Since , we take some such that

Hence, for any , we have (by Theorem 12), and so

that is, . Since is a closed convex cone in , we obtain that

Conversely, we have to prove that

We will prove that the negative polar of is included in ; that is, (by Theorem 12)

Note first that

So consider any . Then, we have

Hence, (since and )

which ensures by Theorem 12 that . Thus, completing the proof.

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 instead of its l.s.c. and the assumption instead of the assumption 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 such that

Corollary 15. Let and Hausdorff topological vector spaces, a set-valued mapping with a closed convex graph, be a bounded convex set in , and . If is calm at with , then one has

Proof. Using the calmness of at and Banach-Alaoglu theorem (see Theorem  3.5 in [21]), we get the weak star compactness of in . Therefore, the assumption with the weak star compactness of ensures the weak star closedness of the cone generated by ; that is, . Thus, the conclusion follows from Theorem 14.

The next results depend on the nonemptiness of the minimum set for the minimal time function defined as follows

We begin with the following lemma.

Lemma 16. Let and be two Hausdorff topological vector spaces, a set-valued mapping with closed convex graph, a bounded convex set in , and with and . For every and for every , one has

This ensures that , for any .

Proof. Let . Then with . First, we prove the inequality

Put , and fix any . Clearly, by definition of the Minkowski function, we can find for any some such that and , and so

which ensures that

Taking finishes the proof of the first inequality.

By the first part, we have whenever and so for any , we can find some , some , and some such that

Then, by convexity of , we have , and so

Hence,

and so by taking , 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 at points outside the graph.

Proposition 17. Assume that the assumptions of Lemma 16 are fulfilled. Then, for every and for every , one has

Proof. By Lemma 16, we have

We turn now to show the inclusion

Fix any . Then

By assumption, we have . Then, there exists ; that is, with . By the first part, we obtain that

and so

since , because . Thus, for any , there exist and such that

and . Thus,

and hence,

Taking yields

Let us prove the reverse inequality

Let be a net in converging to and satisfying

By Lemma 10, we have, for , the following inequality

which ensures that

and so

Taking the limit on , we get that

Using the fact that , we write

Fix any , and take in the previous inequality we get

This ensures that , and hence, the proof is finished.

The following theorem establishes another relationship between the convex subdifferential of and the convex normal cone of at points outside the graph .

Theorem 18. Let and be two Hausdorff topological vector spaces, a set-valued mapping with nonempty closed values of be a bounded convex set in , and . Assume that . Then, one has

If, in addition, is calm at , then one has

Proof. The first inclusion follows directly from the proof of Theorem 14 and Proposition 17. Let us prove the reverse inclusion. Fix any with . First, let us check that . Indeed, by the assumption and Proposition 17, there exists such that

which ensures that . Therefore, by Corollary 15, there exists some and , such that . Using the first part of the proof, we obtain that , and so by combining this equality with the assumption , we get . Thus completing the proof.

5. Application of : Calmness and Exact Penalization

The primary goal in the present section is to make clear that the scalar function 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 , which consists in minimizing the function over all satisfying ,

where is a closed set-valued mapping between two normed vector spaces and and is an extended real-valued function. We begin with the definition of calmness.

Definition 19. Let , , , and be as in the statement of , and consider the following perturbed problem

Let be the feasible set-valued mapping associated with ; that is,

Let and let be a bounded set in . One will say that the problem is calm at with respect to if there exist a constant and such that for every and any one has

The constant and are called the modulus of calmness and neighborhood of calmness for at , respectively.

Remark 20. When is assumed to be normed, the above definition coincides with the definition used in [22, 23] by taking to be the closed unit ball in . Observe that if is calm at with respect to a bounded set , then is necessarily a local solution to .

For any problem , any real number , and any bounded set , we will associate the function defined by

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 in terms of the minimal time function to images associated with the set-valued mapping defining the constraint of the problem.

Theorem 21. Let . If is calm at with respect to with modulus and neighborhood , then is a minimum over of the function , for all . If, in addition, is convex and , the converse holds.

Proof. Let . Given any . Then there exist and such that and . Thus, if , we obtain from the calmness hypothesis that

Fix any . Then by taking in the previous inequality and observing that we obtain that

Hence

Since was taken arbitrarily, the direct implication is established by taking .

Conversely, we assume that is convex and . Fix any and any . We claim first the inequality . Clearly the case is obvious. So we assume that . Since , we have . Then by the finiteness of we can find for any some such that

Hence, there exists such that . Therefore, for any and any , we have

Since is convex, we have , and so , which ensures by the definition of the minimal time function that

Since , we let and . Since is convex and , we have , and hence . Thus, (127) ensures that

After taking , this inequality ensures the desired inequality; that is, . Therefore, for any and any , we have

Hence, is calm at with respect to , and the proof of the theorem is complete.

Acknowledgment

The author extends his appreciations to the Deanship of Scientific Research at King Saud University for funding the work through the research group project no. RGP-VPP-024.

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