Abstract
In this paper, we continue the study of the -prox-regularity that we have started recently for sets. We define an appropriate concept of the -prox-regularity for functions in reflexive smooth Banach spaces by adapting the one given in Hilbert spaces. Our main goal is to study the relationship between the -prox-regularity of a given l.s.c. and the -prox-regularity of its epigraph.
1. Introduction and Preliminaries
Throughout this work, will denote a reflexive smooth Banach space unless otherwise specified. We recall from [1] the concept of -proximal subdifferential.
Definition 1. Let be a lower semicontinuous (l.s.c.) function and , where is finite. We recall that the -proximal subdifferential of at is defined as if and only if there exists such that
Here, is the normalised duality mapping on , and is the functional defined from to by
The -proximal normal cone of a nonempty closed subset in at is defined as the -proximal subdifferential of the indicator function of , that is, .
Another proximal subdifferential is defined (see [5]) geometrically via the -proximal normal cone of the epigraph as follows:
It has been proved in [2] that in general, we have the inclusion . The generalized projection on a closed nonempty set is defined as follows: if and only if (see [3] for convex sets and see [1] for nonconvex sets). We also recall (see for instance [2]) the definition of the Fréchet subdifferential and Fréchet normal cone as follows: if and only if for all , there exists such that
The Fréchet normal cone of a nonempty closed subset in at is defined as .
2. Generalized -Prox-Regular Sets
In this section, we recall the concept of the generalized -prox-regularity of sets introduced and studied in [4] and other results needed in our work.
Definition 2. Let be a nonempty closed set in a reflexive Banach space and let . We will say that is generalized V-prox-regular at if and only if there exist and such that for all and for any , the point is a generalized projection of on , that is, .
We quote from [5, 6] the following two results.
Lemma 3. For any and any (i.e., ), we have
Proposition 4. Let be an Asplund space and let be a proper l.s.c. function around . Then, for any with , there exist sequences with , , and such that .
We recall from [2] that is -proximal trustworthy provided that for any , any two functions and any such that is lower semicontinuous and is Lipschitz around the following fuzzy sum rule holds:
Here, , and denotes the closed unit ball in . The proof of the following proposition can be found in [2].
Proposition 5. Let be a -proximal trustworthy space. Let be a closed subset of with and let . Then, for any , there exists such that .
3. -Prox-Regular Functions
In Hilbert spaces setting, the authors in [7] extended and studied the concept of prox-regularity for functions which has been introduced for finite dimensional spaces in [8]. Many papers studied this concept in finite dimensional spaces and its applications to nonsmooth optimization (see for instance [9] and the references therein). Our aim here is to extend the concept of prox-regularity to reflexive smooth Banach spaces by using the concept of the -proximal normal cone and the functional . Another way to extend this concept has been proposed in [10].
Definition 6. Assume that is a reflexive smooth Banach space. A l.s.c. function is said to be a -prox-regular at for if and only if there exist such that with , and such that
We say that is -prox-regular at if it is -prox-regular for any . Obviously, this concept coincides with the one studied in [7] in Hilbert spaces. Using this definition for functions, we can associate to it another definition of the -prox-regularity for sets via the indicator function. We have the following definition:
Definition 7. Let be a reflexive smooth Banach space. A closed nonempty set is said to be -prox-regular at for if and only if the indicator function is -prox-regular at for .
Proposition 8. Let be a reflexive smooth Banach space. A closed set is generalized -prox-regular at in the sense of Definition 2, if and only if the indicator function is -prox-regular at for in the sense of Definition 6.
Proof. Assume that is generalized -prox-regular at , then there exist and such that for all and for any , the point is a generalized projection of on , that is, . By definition of generalized projection, we have
So, for any , we have
Thus, for any . Hence, and so
On the other hand, we have which implies that with and . Thus, the inequality (12) ensures that is -prox-regular at for . For the opposite direction, we follow the same lines as in the direct one.
In the next theorem, we study the relationship between the -prox-regularity of a function and the -prox-regularity of its epigraph.
Theorem 9. Let be a function that is l.s.c. at . If is -prox-regular at for , then the epigraph is -prox-regular at for. If, in addition, the space is -uniformly convex, then the previous implication becomes an equivalence, that is, is -prox-regular at for if and only if the epigraph is -prox-regular at for .
Proof. Assume that is -prox-regular at for , fix and such that with , and such that
By l.s.c. of , we choose with and such that for any , we have . Fix any , with and with . Clearly, , which ensures that and so Take now any with . Notice that which yields by (13) and hence,
On the other hand, we have So,
Thus, for any , with and with , we have for any with . This means that the epigraph is -prox-regular at for .
Assume now that the space is a -uniformly convex space. Assume that the epigraph is -prox-regular at for , choose and such that for any and any with and with , we have for all with . Using the fact that space is a -uniformly convex space, we can find a positive constant depending only on the space and the constant such that
Choose now a positive number such that and such that (according to the l.s.c. of )
Here, depends only on the space and . Then, for any , we have and so
Also, we have by (23) one that has and hence since (by (23) again), we may fix some positive number such that
Take now any with and and with . Then, applying (20) with instead of , instead of , and instead of , we obtain which is equivalent to (with ) and so
Note that
Thus,
Now, we develop the expression
Observe that
Thus,
But as , one has
Using the choice of in the inequality (26), we have which ensure, respectively,
Since , we obtain
Therefore, it follows from (35) that and so which ensures that
Thus, by (31) and (35), we obtain
Now, it remains to show the conclusion for satisfying . First choose, by the l.s.c. of , some positive number such that
Now, fix any with only , and fix any . Thus, if , then
But since (by (43)), we get only
Therefore,
Observe that and hence, we get
Finally, the relation (41) together with the last inequality conclude that is -prox-regular at for .
Remark 10. In the previous theorem, we proved the following: if is -prox-regular at for , then we have the -prox-regularity of at for . Unfortunately, we do not obtain the generalized -prox-regularity of the epigraph in the sense of Definition 2 since is a generalized -prox-regularity at if and only if it is -prox-regularity at for which cannot be the case (since ) under the -prox-regularity of at . To ensure the generalized -prox-regularity of the epigraph, we need another kind of regularity called the -primal lower nice function introduced and studied in [5].
Definition 11. Let be a l.s.c. function. We will say that is -primal lower nice at , if there exist such that whenever , , and . Here,
This concept is stronger than the -prox-regularity. Indeed, as mentioned in [8] for finite dimensional spaces, obviously, any function that is a -p.l.n. function is -prox-regular. However, the inverse is not true. For a -p.l.n function, the condition (49) must hold for all -proximal gradients with the linear growth condition, whereas for -prox-regular functions, the condition (7) only has to hold for -proximal subgradients close to a fixed and just in a neighborhood of with close to . For instance (see [8]), the function on with for and for is easily seen to be -prox-regular at for but not -p.l.n. for this . We have to notice that this definition of -p.l.n. functions extends, to reflexive smooth Banach spaces, the definition of primal lower nice functions defined in finite dimension spaces in [11], and in Hilbert spaces in [12]. Our next proposition shows the generalized -prox-regularity of the epigraph in the sense of Definition 2 of any -primal lower nice functions.
Proposition 12. Let be a -proximal trustworthy space. If is -primal lower nice at , then is generalized -prox-regular at .
Proof. By definition of -p.l.n. at , we have positive numbers such that for any , any , and any , we have
Take and and with , where is taken in such that
Fix now any with and . Clearly, .
Case 1. . In this case, we necessarily have . Then, , so and for every . Hence, by (3.18), we obtain which entails
Since , we have , and we have and and hence (54) entails with
Hence,
So,
Dividing by yields
On the other hand, we have
So,
Thus, (58) becomes and so
Observe that
Therefore, for any that is,
Case 2. . In this case, we have and so by Lemma 3, we obtain . Using the fact that , we get and so by Proposition 4, there exists a sequence with , , and (i.e., such that . Using now Proposition 5, we choose for each
This ensures that and so . Consequently, with
Also, we have
Let Then, that is, . For large enough, (i.e., ), we have and . Assume for a moment that . Let , , we see that and hence by definition of -p.l.n. functions with and , we obtain
Multiplying this inequality by , we get
Let . Clearly, and . So, for any , we have
Now, taking the limit as that yields by continuity of and and so
Thus,
Dividing by gives and so
This ensures for for any that is,
Finally, we obtained from Case 1 and Case 2 two positive numbers and such that for any and and any and , we have
This means by Definition 2 that the epigraph is generalized -prox-regular at .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Acknowledgments
The authors extend their appreciations to the Deanship of Scientific Research at King Saud University for funding the work through the research group project no. RGP-024. The authors would like to thank the referee for the complete reading of the first version of this work and for the suggestions allowing us to improve the presentation of the paper.