Abstract
In 2010, Bounkhel et al. introduced new proximal concepts (analytic proximal subdifferential, geometric proximal subdifferential, and proximal normal cone) in reflexive smooth Banach spaces. They proved, in -uniformly convex and -uniformly smooth Banach spaces, the density theorem for the new concepts of proximal subdifferential and various important properties for both proximal subdifferential concepts and the proximal normal cone concept. In this paper, we establish calculus rules (fuzzy sum rule and chain rule) for both proximal subdifferentials and we prove the Bishop-Phelps theorem for the proximal normal cone. The limiting concept for both proximal subdifferentials and for the proximal normal cone is defined and studied. We prove that both limiting constructions coincide with the Mordukhovich constructions under some assumptions on the space. Applications to nonconvex minimisation problems and nonconvex variational inequalities are established.
1. Introduction
In [1], the authors introduced the concept of geometric proximal subdifferential and the concept of analytic proximal subdifferential in reflexive Banach spaces for . They also introduced the concept of generalised proximal normal cone for . The main tool in their approach is the generalised projection operator and the functional (see [1, 2]). They proved various important properties for their concepts.
In order to avoid any confusion with other existing proximal subdifferential and proximal normal cones in Banach spaces (see, e.g., [3–6]) we will call , , and analytic -proximal subdifferential, geometric -proximal subdifferential, and -proximal normal cone, respectively. These names are more natural since these concepts are strongly based on and we use these names in all the paper.
The main aim of this work is to prove the fuzzy sum rule and the chain rule for both proximal subdifferentials. Our results extend the existing results on the usual proximal subdifferential from Hilbert spaces setting to our Banach spaces setting. We start by proving that whenever the space admits an equivalent norm which is away from the origin. Sections 3 and 4 are devoted to proving the fuzzy sum rule and the chain rule. In Section 5, we define the limiting constructions for the -proximal concepts (the analytic and geometric -proximal subdifferential and the -proximal normal cone). In this section we prove that these limiting constructions coincide with the limiting constructions in [7] whenever the space admits an equivalent norm with a power of that norm being away from the origin. Section 5 is ended by proving the well known Bishop-Phelps theorem for the -proximal normal cone and its limiting construction. The paper is closed by an application, in which, we derive a necessary optimality condition for nonconvex minimisation problems and nonconvex variational inequalities in terms of generalised projections.
2. Preliminaries
Let be a reflexive smooth Banach space with topological dual space . We will denote by and the closed unit ball centred at the origin in and , respectively. Let be a function and where is finite. We recall from [1] the concept of analytic -proximal subdifferential (called in [1] analytic proximal subdifferential). An element belongs to provided that there exists so thatfor very close to , where is the normalized duality mapping and is a functional defined by
For a closed nonempty set in and , the authors in [1] defined the concept of -proximal normal cone (called in [1] proximal normal cone) by . Using this -proximal normal cone, the authors in [1] defined another concept of -proximal subdifferential, called geometric proximal subdifferential, as follows: An element belongs to provided , where is the epigraph of defined by . In this paper will be called geometric -proximal subdifferential. We recall, respectively, the well known concepts of proximal subdifferential and Fréchet subdifferential (see, e.g., [5]):(i) if and only if there exist such that(ii) if and only if for any there exists such thatThe Fréchet and proximal normal cone are defined as and Notice that (see [7, 8]) Fréchet and proximal subdifferential can be defined geometrically by the formulas and .
We recall (see, e.g., [1, 9]) the definition of -uniformly convex and -uniformly smooth Banach spaces. The space is said to be -uniformly convex (resp., -uniformly smooth) if there is a constant such that where and are defined, respectively, by Notice that the constants and in the previous definition always satisfy and .
The following inclusions have been proved in [1]:Consider the following two assumptions on the norm on at a given point : (A1), for close to and for some and .(A2), for close to and for some and .Using Lemma in [1], we get that (A1) is satisfied for any -uniformly smooth Banach spaces and (A2) is satisfied for any -uniformly convex Banach spaces. Easily we can check that if (A1) is satisfied with , then we have and and so and . This is the case, for instance, of spaces with . The reverse inclusions and hold whenever (A2) is satisfied with and so and which is the case of spaces with . In the general case we cannot provide a relationship between and the two -proximal subdifferentials and .
We recall from [1] the following important characterisation of -proximal normal cone in terms of the generalised projection.
Proposition 1. For any closed set and any point we have
Here is the generalised projection operator defined as follows (see [10, 11]):
We summarise in the following proposition some needed results of the analytic and geometric -proximal subdifferentials proved in [1].
Proposition 2. Let be a -uniformly convex and -uniformly smooth Banach space and let be a lower semicontinuous function and :(i)If is on a neighbourhood of , then (ii)Let , and . Then
It has been proved in [1] that the equality form holds for convex l.s.c. functions and for indicator functions associated with closed nonempty sets in any reflexive smooth Banach space. So, it is a natural question to ask whether this equality is valid for general forms of functions. In the following proposition we give a positive answer with additional assumptions on the space for any l.s.c. function.
Proposition 3. Let be -uniformly convex and -uniformly smooth Banach space. Assume that admits an equivalent which is on . Let be the functional associated with . Let be a lower semicontinuous function at . Then .
Proof. The direct inclusion has been proved in [1]. Let ; that is, and hence there exists such that It follows that By lower semicontinuity of we may choose some such that , for all . Also, we can take too small so that Therefore, for any we have which yieldsSet and . Then , for all ; that is, is a local minimum of and so . Now, we can easily check that is around with for very close to . Indeed, since we can find smaller than for which we have , for any . Thus by the differentiability of the equivalent norm we have for any that is, the function is differentiable at any and . Now, we use the property of the norm around , that is, the property of the duality mapping around , and we obtain the property of around with . Now, we return to and we use the previous proposition to get ; that is, and hence the proof is complete.
Remark 4. The assumption on the norm used in the previous proposition is true for any Hilbert space and for many other Banach spaces (see, e.g., [9, 12]). Using Theorem in Section in [9] we get that all spaces for satisfy the property with their canonical norm and the same result is also true for the Sobolev spaces (). We refer the reader for more examples and discussions to [9, 12] and the references therein.
3. Fuzzy Sum Rules
In this section, we are going to prove the fuzzy sum rule for the analytic -proximal subdifferential. To do that, we need the following new version of Borwein-Preiss variational principle in terms of the functional . For its proof we refer the reader to [13].
Theorem 5. Let be -uniformly convex and -uniformly smooth Banach space and let be bounded below lower semicontinuous function, and . Suppose that is a point satisfying . Then for any there exist points and withand having the property that the function has a unique minimum at .
Let us prove the following lemma needed in the proof of the main result in this section. Its proof follows the same lines of the proof of Proposition in [8].
Lemma 6. Let be -uniformly convex and -uniformly smooth Banach space. Let , , be l.s.c. functions at ; one of them is Lipschitz continuous around . Let be a closed convex subset of with . Let and let be any equivalent norm on . Then for any with we have
Proof. Obviously we always have and so Conversely, let such that Since and are bounded below on we get This ensures that for some which yields that since .
We use now the Lipschitz continuity of over for some to getTaking the limit as yields the desired inequality and hence the proof is finished.
Now, we assume that the space satisfies an additional assumption. Assume that admits an equivalent such that (for some ) is -differentiable on . We refer the reader to [9, 12] for many examples of Banach space satisfying this assumption. So, we are ready to prove the main result.
Theorem 7. Let be -uniformly smooth and -uniformly convex Banach space. Let , , be l.s.c. functions at ; one of them is Lipschitz continuous around . Assume that admits an equivalent such that (for some ) is -differentiable on and let be the functional associated with that . Let . Then for any , there exist with , such that
Proof. Let . There exist such that attains a minimum over at . We take also small enough so that and are bounded below on and is Lipschitz on which is satisfied for any . Let where . Now consider the problem of minimising over . Applying Lemma 6 with the functions and yields the conclusion that the nonnegative quantity as ; that is, for large enough the couple satisfies We apply now the variational principle stated in Theorem 5 with , , and . Then there exist two points and such thatand such that the function has a unique minimum at over the space . Thus for sufficiently large (since ) both and are in , so that the fact of the minimisation implies the separate necessary conditions:Thus using the fact that the functions and are on with and by using Proposition 2, we can write Adding these two inclusions yields Let . For large enough we can ensure that and and so Thus, it remains to show that , , for large enough. Since , we haveand so for large enough. From this inequality we derive Using now the l.s.c. of both functions and and and we obtainand so we deduce from the previous inequalities that Using this equality and the l.s.c. of we getThis inequality with the l.s.c. of ensures that Similarly for we obtain These two equalities ensure , , for large enough.
We notice that the fuzzy sum rule for the geometric -proximal subdifferential cannot be deduced directly from the previous one. However, it will be used for that purpose. First we need to prove the following proposition.
Proposition 8. Let be -uniformly smooth and -uniformly convex Banach space. Let be a l.s.c. function at . Assume that admits an equivalent such that (for some ) is -differentiable on and let be the functional associated with that . For any and any , there exist with such that
Proof. Let . Fix any . Then by definition of the Fréchet subdifferential there exists such that attains a minimum over at . Hence Using the fuzzy sum rule for the analytic -proximal subdifferential in Theorem 7 we can find with such thatThus,and so the proof of the proposition is complete.
Now, we are ready to prove the fuzzy sum rule for the geometric -proximal subdifferential.
Theorem 9. Let be -uniformly smooth and -uniformly convex Banach space. Let , , be l.s.c. functions at ; one of them is Lipschitz continuous around . Assume that admits an equivalent such that (for some ) is -differentiable on and let be the functional associated with that . If , then for any , there exist with , , such that
Proof. Fix any and let be the Lipschitz constant of around . Let . Assume that . By Proposition 2 we get . Then, by Proposition 8 there exists with , such that We use now the fuzzy sum rule for the analytic -proximal subdifferential to obtain some points with , , such that Obviously we have . Also we have This completes the proof.
The following corollary is a direct consequence of Theorem 7. It is also valid for the geometric -proximal subdifferential.
Corollary 10. Let be -uniformly smooth and -uniformly convex Banach space. Assume that admits an equivalent such that (for some ) is -differentiable on and let be the functional associated with that . Let be locally Lipschitz function around . If minimises over , then for any , there exist with and such that
4. Chain Rule
Using the fuzzy sum rule established in the previous section for the analytic -proximal subdifferential, we prove the chain rule for both analytic and geometric -proximal subdifferentials. Let be another -uniformly convex and -uniformly smooth Banach space and let and . In this section, will denote the norm on and the closed unit ball in will be denoted by . We will denote by functional (2) associated with the . We will say that is a locally -Lipschitz mapping around provided that there exist and such that
Theorem 11. Let , be a locally -Lipschitz mapping around , and let be locally Lipschitz around . Assume that (resp., ) admit an equivalent (resp., ) such that (resp., ) (for some ) is -differentiable on (resp., ) and let (resp., ) be the functional associated with (resp., ). Let . Then for all there exist , , and such that and
Proof. Fix . By Lipschitz property of around and by definition of the analytic -proximal subdifferential there exist a positive number (we may assume that ), , and such thatSet , the graph of in . The previous inequalities ensure that for any This means that the function attains a local minimum at and hence Applying our fuzzy sum rule in Theorem 7 with there exist and in such that for some we have This ensures that ; that is, . Using Proposition 2 we obtain for some Thus Hence there exist , , , and such that for any we haveThe last inequality follows from Lemma in [1] and hence for any we have and hence we getwhere Thus We use the fuzzy sum rule once again to obtain two points and in such thatThe Lipschitz property of with ratio around ensures that is Lipschitz with ratio around and hence by Proposition 2 we obtain Also the uniform continuity of ensures by shrinking that and hence Therefore, by taking and , we get and with and , thus completing the proof.
Remark 12. As for the fuzzy sum rule for geometric -proximal subdifferential, we combine the chain rule for the analytic -proximal subdifferential and Proposition 8 to prove easily the chain rule for the geometric -proximal subdifferential.
5. Limiting -Proximal Subdifferential and Limiting -Proximal Normal Cone
After proving the density theorem for in [1] and consequently for , it is a natural question to consider the limiting object of both -proximal subdifferential concepts as follows:where means with . These two subdifferentials will be called, respectively, the limting analytic -proximal subdifferential and the limiting geometric -proximal subdifferential. Our main aim in this section is to prove, under the assumptions of Theorem 7, that these two constructions coincide with the limiting Fréchet subdifferential (also called Mordukhovich subdifferential) (see [7]).
Theorem 13. Let be -uniformly smooth and -uniformly convex Banach space. Let be a l.s.c. function at . Assume that admits an equivalent such that (for some ) is -differentiable on and let be the functional associated with that . ThenHere and are limiting Fréchet subdifferential and limiting proximal subdifferential, respectively; that is, and
Proof. The last equality has been proved in [6] which is our setting. So, it remains to prove the other two equalities. It will be enough to prove the inclusion . Fix . There exists with such that , . Using Proposition 8 with we get a sequence with such that . So, there exists such that . Clearly and . Thus, and hence the proof is complete.
As a first consequence of this result we have the following theorem, which is a new result in Banach spaces.
Theorem 14. Let be -uniformly smooth and -uniformly convex Banach space. Let be any closed nonempty set of . Assume that admits an equivalent such that (for some ) is -differentiable on and let be the functional associated with that . Then
We establish in the following theorem a new characterisation in terms of analytic and geometric -proximal subdifferentials of the singular subdifferential defined (see [7]) by
Theorem 15. Let be -uniformly smooth and -uniformly convex Banach space. Let be a l.s.c. function at . Assume that admits an equivalent such that (for some ) is -differentiable on and let be the functional associated with that . Then
Proof. The first equality is proved in [7]. To conclude, it is enough to prove the inclusion Fix any ; that is, . Then there exist and such that Clearly , for all . We distinguish two cases.
Case 1. for sufficiently large. For this case we have ; that is, . Using Proposition 8 we get a sequence with such that ; that is, for some . Let and . Then with and .
Case 2. for some subsequence of . In this case we get subsequences of , , (still denoted by the same notation) for which we have . Then we use Theorem in [6] to get sequences , , such that , , and such that . Hence with and . Similarly to the first case, by using Proposition 8 we get a sequence such that with and . Then for some we have with and .
Thus, from both cases we get and so the proof is complete.
We close this section by proving the well known Bishop-Phelps density theorem in terms of -proximal normal cone in smooth Banach spaces (see, e.g., [7]). To do that we need the following recent result on the generalised projection on nonconvex closed sets [13]. It is an analogue result to Lau’s theorem for metric projections in reflexive Banach spaces ([14]).
Theorem 16. Let be -uniformly convex and -uniformly smooth Banach space, and let be any closed nonempty set of . Then there is a dense set of points in admitting unique generalised projection on ; that is, for any , there exists with , .
Now we are ready to prove the following result.
Theorem 17. Let be -uniformly convex and -uniformly smooth Banach space and let be a nonempty closed subset of . Then the set of points with is dense in the boundary of .
Proof. Fix any and . There exist such that . Using Theorem 16 we can find with and . Let ; that is, and Then by definition of the -proximal normal cone we get . Obviously since and , which ensures Let and take . From what precedes we can find for any a sequence , , with and . We are going to prove that . By the -uniform smoothness of the space , there exists by Lemma in [1] two positive numbers and such thatThen for large enough we can write Here is the normalized duality mapping on . Therefore, since we have This ensures that as . Hence the sequence is bounded by some (since we have ) and soWe use now the -uniform convexity of to obtain a constant (depending on , , and ) such that Thus, This ensures that , thus completing the proof.
In the next theorem we establish an analogue result to the previous theorem with the limiting -proximal normal cone. First, we need to define the following concept of normal compactness with respect to the -proximal normal cone.
Definition 18. A closed set is said to be normally compact around if there exist positive numbers , and a compact subset of such that
Clearly, this normal compactness with respect to -proximal normal cone is ensured by the normal compactness with respect to the Fréchet normal cone introduced in [15] and used by many authors (see, e.g., [7]). We point out that using Theorem 13 the next result is covered by Corollary in [7] provided that the space satisfies the assumptions in Theorem 13 and the normal compactness is satisfied with the Fréchet normal cone. However, our result is proved for any -uniformly convex and -uniformly smooth Banach space and under the normal compactness relatively to the -proximal normal cone.
Theorem 19. Let be -uniformly convex and -uniformly smooth Banach space and let be a nonempty closed subset of . Assume that is normally compact around with respect to the -proximal normal cone. If , then .
Proof. Let . Then by the previous proposition there exists a sequence with . Let with . Set . By extracting a subsequence we may assume that the sequence weakly converges to some limit . Obviously . It remains to prove that . Assume by contradiction that ; that is, the sequence and so , . Using the normal compactness of we get and so which contradicts the fact that . Thus and so . This completes the proof.
We use this theorem to prove a useful result for the nonemptiness of the limiting geometric -proximal subdifferential for locally Lipschitz functions and consequently the nonemptiness of the limiting geometric -proximal subdifferential.
Theorem 20. Let be -uniformly convex and -uniformly smooth Banach space and let be a nonempty closed subset of . Let be a locally Lipschitz around . Then
Proof. Let , which is obviously closed and epi-Lipschitz around due to the Lipschitz continuity of around and so is normally compact with respect to the -proximal normal cone (see Proposition in [15]). Also, obviously is a boundary point of . Using the previous proposition we get . Then there exist and with . Notice first that obviously we have and . In order to prove that , we assume by contradiction that . Then and so which means that and so Corollary in [7] ensures by the Lipschitz continuity of that and this contradicts the fact that . So this case is not possible. Consequently, we have . Then ; that is, and so and so is nonempty and the proof is complete.
Remark 21. The nonemptiness of the limiting analytic -proximal subdifferential for locally Lipschitz functions in -uniformly convex and -uniformly smooth Banach space cannot be deduced from the previous result in Theorem 20. However, the nonemptiness of is ensured under the assumptions of Theorem 7.
6. Applications
Throughout this section the space is -uniformly convex and -uniformly smooth Banach space admitting an equivalent such that (for some ) is -differentiable on and the functional is associated with that . Our objective in this section is to present two applications of the results proved in the previous sections. Our results in this section are new in the Banach spaces setting. We establish in terms of the generalised projection some necessary optimality conditions to nonconvex minimisation problems and nonconvex variational inequalities.
6.1. Necessary Optimality Conditions to Nonconvex Minimisation Problems
Consider the following constrained minimisation problem: where is a l.s.c. function and is a closed nonempty set in .
Proposition 22. Assume that is locally Lipschitz at . If is a solution of , then there exist , , , and such that .
Proof. Assume that is a solution of . Then is a global minimum of the function over and so . Using the exact sum rule for the limiting Fréchet subdifferential (see, e.g., [7]) we get . Then there exists such that . Using Theorem 14 we obtain , , and such that and hence the proof is complete.
A direct consequence of this proposition is given in the following corollary.
Corollary 23. Assume that is a -proximal normally regular point of ; that is, and assume that is locally Lipschitz at . If is a solution of , then , for some and for some .
Proof. Following the same lines of the proof of Proposition 22, we obtain some such that . Since is a -proximal normally regular point of , that is, we have by Theorem 13 the equality and so . Consequently, by the characterisation of the -proximal normal cone given in Proposition 1 there exists such that and hence the proof is complete.
6.2. Necessary Optimality Conditions to Nonconvex Variational Inequalities
Let be an arbitrary mapping, let be a nonempty closed set not necessarily convex, and let be a bifunction satisfying on . Consider the following nonconvex variational inequality:Our aim is to derive necessary conditions for a solution to (86) in terms of the generalised projection on . First, we observe that this variational inequality coincides with the one studied in [16] when is convex and is taken to satisfy on .
Proposition 24. Assume that is a solution of (86) and that is locally Lipschitz at . Then there exists some and sequences , , and such that , , and and satisfying is a fixed point of the set-valued mapping , for any .
Proof. Assume that is a solution of (86). Then is a global minimum of the function . Hence Using the exact sum rule for the Fréchet limiting subdifferential (see, e.g., [7]) and the fact that is locally Lipschitz at , we obtainThen there exists some such that By the characterisation given in Theorem 14, there exist , , and such that . Let . Then and and hence is a fixed point of the set-valued mapping , for any and hence the proof is complete.
The following corollary is a direct consequence of the previous proposition at -proximal normally regular points.
Corollary 25. Assume that is a -proximal normally regular point of and that is a solution of (86). Assume that is locally Lipschitz at . Then there exist and such that is a fixed point of the set-valued mapping .
Competing Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author extends his appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group Project no. RGP-024.