Iterative Schemes for Nonconvex Quasi-Variational Problems with -Prox-Regular Data in Banach Spaces

Abstract

In this paper, we propose an extension of quasi-equilibrium problems from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. The proposed problem is called quasi-variational problem. We study the convergence of some algorithms to solutions of the proposed nonconvex problems in Banach spaces.

1. Introduction

Let be a Banach space and let be the dual space of . Let denote the duality pairing of and . Let be a set-valued mapping with nonempty closed values and let be a bifunction satisfying for all . We associate with a closed convex valued set-valued mapping and a convex bifunction the following well known quasi-equilibrium problem:In this paper we propose the following appropriate extensions of from the convex case to the nonconvex case in Banach spaces setting. We associate with and the following nonconvex quasi-variational problem equilibrium problems:where (resp. ) is the -proximal subdifferential (resp. -proximal normal cone) introduced and studied in [1].

The proposed nonconvex quasi-variational problem extends many existing quasi-equilibrium problems and quasi-variational inequalities from the convex case to the nonconvex case and from Hilbert spaces setting to Banach spaces setting.(1)If is a Hilbert space, the proposed becomes  where and are the usual proximal subdifferential and proximal normal cone in Hilbert spaces. This problem has been introduced and studied in Bounkhel et al. [2]. Since then it has been studied and extended in various ways in Hilbert spaces by the authors in [3] and in Noor [4] and many works (see for instances Noor et al. [5, 6]).(2)If is a Hilbert space, is a convex closed set in , is a convex bifunction, and , then becomes the following well known convex equilibrium problem:  which has been studied in various works (see for instance Moudafi [7], M. A. Noor and K. I. Noor [5], and the references therein).(3)If , with , is a nonlinear operator then reduces to  which will be shown in Section 4 to be equivalent in the uniform -prox-regular case, for some , to the following quasi-variational inequality:  This inequality is new in Banach spaces. However, it has been studied, in Hilbert spaces, in Bounkhel et al. [2], when is a uniformly -prox-regular set (see also Bounkhel and Al-Sinan [8] and Noor et al. [5, 6]). When and the last inequality becomes  which is known as the classical variational inequality introduced and studied in Stampacchia [9].Our main objective of the present paper is to prove the convergence of some algorithms to solutions of the proposed nonconvex quasi-variational problem .

2. Preliminaries

In order to prepare the framework of our study we need to state some concepts and results. First we recall (see for instance [1, 10]) 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 . Also we need to recall from [1] the concept of -proximal subdifferential (called in [1] generalised proximal subdifferential). An element belongs to provided that there exists so thatfor very close to , where is the normalised 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] generalised proximal normal cone) by , where denotes the indicator function associated with , that is, if and if . We recall, respectively, the concepts of limiting Fréchet subdifferential and limiting -proximal subdifferential (see [11]): where means with and The limiting Fréchet normal cone is defined similarly, that is,where denotes with and is the Fréchet normal cone which is defined by .

These all nonconvex objects coincide with their analogues defined in Convex Analysis whenever the data are convex as the following proposition shows (see [1]).

Proposition 1. Let be a reflexive Banach space. (1)Let be a l.s.c. convex function and with . Then (2)Let be a closed convex subset in of and . Then

The following result is needed in our study. It has been proved in [11].

Theorem 2. Let be a -uniformly smooth and -uniformly convex Banach space. Assume that admits an equivalent norm such that (for some ) is -differentiable on and let be the functional associated with that norm . (1)Let , be a l.s.c. function at . Then(2)Let be any closed nonempty set of . Then

We notice that the class of spaces satisfying the assumptions of the previous theorem is very large; it contains obviously any Hilbert space and spaces and Sobolev spaces with (see Theorem 1.1 in Section 5 in [10, 12]) and for more examples and discussions we refer to [10, 12]. We close this section with the following two concepts of uniform -prox-regularity for functions and sets (see [13]).

Definition 3. Let be a reflexive smooth Banach space. For a given , a subset is -uniformly prox-regular with respect to provided that for all and all nonzero we haveWe use the convention for .

Obviously, this class contains the class of uniformly prox-regular sets ([14, 15]) from Hilbert spaces to Banach spaces since in Hilbert spaces we have and the -proximal normal cone coincides with the usual proximal normal cone .

Definition 4. Let be a reflexive smooth Banach space. Let be a l.s.c. function and let be a nonempty closed set in . We recall from [13] that is said to be uniformly -prox-regular over provided that for all and all we haveWe say that is uniformly -prox-regular around provided that is uniformly -prox-regular over some closed neighborhood of ; that is, there exists a closed neighborhood of such that the inequality (18) holds for any .

The following example is quoted from [13]. For its proof we refer the reader to [13].

Example 5. (1) Any l.s.c. proper convex function is uniformly -prox-regular over any nonempty closed set in its domain with .
(2) Both the indicator function and the distance function of uniformly -prox-regular set are uniformly -prox-regular over with respect to the same constant .
(3) Any lower- function over convex strongly compact in is uniformly -prox-regular over with some (see [13] for the definition of lower- functions).

The following two lemmas are needed in our proofs in Section 4. The proof of the first one is proved in [1]. The second one is proved in [16].

Lemma 6. Let be a -uniformly convex and -uniformly smooth Banach space and be a bounded set. Then for some we have

Lemma 7. If is a uniformly convex Banach space, then the inequality holds for all and in , where .

3. Main Results

First we show that in the convex case coincides with the quasi-equilibrium problem .

Proposition 8. Let be a reflexive Banach space. Assume that is a closed convex set-valued mapping and is a convex bifunction satisfying for any . Then we have     .

Proof. ⁡
. Let be a solution of ; that is, there exists such that . Since is a closed convex set, the -proximal normal cone coincides with the convex normal cone (by Proposition 1) and so On the other hand, the convexity of the bifunction and Proposition 1 yield Since we have (by assumption) and hence the previous two inequalities ensure that is, is a solution of .
. Let be a solution of , that is, Since is a closed convex set and is a convex function, the function admits at a global minimum on . It follows that which is equivalent to and hence the proof is complete since and .

In the next proposition we establish an inequality characterisation of the proposed nonconvex quasi-variational problem whenever the data and are uniformly -prox-regular.

Proposition 9. Let be a reflexive Banach space and . Assume that is uniformly -prox-regular with ratio and that is uniformly -prox-regular over with ratio . Assume also that is -Lipschitz around and for any . If is a solution of , then is a solution of the following nonconvex quasi-equilibrium problem. Find such that for some nonnegative .

Proof. Assume that is a solution of ; that is, such that . By uniform -prox-regularity of the set we have The -Lipschitz continuity of ensures that and so we obtainOn the other hand the uniform -prox-regularity of over with ratio ; we haveCombining this inequality (27) with (26) we obtainSince we have and so (28) becomes with . Thus the proof is complete.

It is a natural question to ask whether the converse in the previous proposition is true or not. The answer is positive provided that the space and the data and satisfy some additional assumptions as the following proposition shows.

Proposition 10. Let be a -uniformly smooth and -uniformly convex Banach space. Assume that admits an equivalent norm such that (for some ) is -differentiable on and let be the functional associated with that norm . Assume that is -proximal normally regular at , that is, and that is -proximal subdifferentially regular at , that is, . Assume that for any . If is a solution of for some , then is a solution of .

Proof. Let be a solution of for some ; that is, Then is a global minimum of the function over and hence Note that the function is differentiable and its gradient is given by . Using the fact that the limiting -proximal subdifferential coincides with the limiting Fréchet subdifferential (by Theorem 2) and the exact sum rules for the limiting Fréchet subdifferential (see for instance [17]) we can writeThis is equivalent to say that . Thus completing the proof since and .

The following proposition has its own interest and is needed to prove the equivalence between and whenever and are uniformly -prox-regular.

Proposition 11. Let be a reflexive Banach space and let be a l.s.c. function and let . If is uniformly -prox-regular around , then ; that is, is -proximal subdifferentially regular at . Consequently, for any uniformly -prox-regular closed set at we have ; that is, is -proximal normally regular at .

Proof. We only prove the first assertion; the second one follows directly from the first one and Example 5 Part (2). Since we always have the inclusion , it is enough to prove the reverse one, that is, . Let ; that is, there exists and such that . By the uniform -prox-regularity of around , there exist and such that for any and any Since we can write for large enough that and hence by (33) we haveFix any . Clearly and hence (34) ensures Using now the fact that , the continuity of and , and the weak convergence of to to pass to the limit as goes to and to get for any , this means by definition that and the proof is complete.

Using this result together with Propositions 9 and 10 we obtain the equivalence between and .

Proposition 12. Let be a -uniformly smooth and -uniformly convex Banach space and . Assume that admits an equivalent norm such that (for some ) is -differentiable on and let be the functional associated with that norm . Assume that is uniformly -prox-regular with ratio and that is uniformly -prox-regular over with ratio . Assume also that is -Lipschitz around and for any . Then is equivalent to for some .

4. Convergence Analysis

4.1. Case 1: Is a Constant Set-Valued Mapping

In this case the proposed problem becomes as follows: In this subsection we propose the following algorithm.

Algorithm 13. Let and for all ; (1)Select ;(2)For select such that

Theorem 14. Let be a -uniformly convex Banach space. Let be a closed nonempty subset of and let be a bifunction satisfying for any . Let be a sequence generated by Algorithm 13. Assume that (1) is -uniformly prox-regular with some ;(2) is ball compact; that is, is compact for any ;(3)The solution set of is nonempty;(4) is -strongly monotone over for some ; that is,  where ;(5) is upper semicontinuous with respect to the first variable over ; that is, (6)The bifurcation is -Lipschitz with respect to the second variable and is -uniformly prox-regular over with some ;(7)There exists such that for all ;(8)The parameters satisfy the inequalities .Then, there exists subsequence of converges to which solves .

Proof. Let be a solution of . Then by Proposition 9 we have for . By the -strong monotonicity of over we have By setting in these two inequalities we get Combining these two inequalities we obtain Using the 8th assumption in Theorem 14 we have and hence This combined with Algorithm 13 giveswith . Therefore,Define now a sequence of nonnegative real numbers . It is not hard to verify thatIndeed,It follows that which ensures with (46) that Using the assumption in the 8th assumption we obtain Therefore, the sequence is a nonincreasing converging sequence to some limit and so it is bounded by some . Thus by the properties of the functional we obtain and so that is, is bounded and so by the -uniform convexity of (by Lemma 6) we have for some depending on and on the space the inequality where is the normalised duality mapping on and is the functional defined by Using now (46) and (47) and the assumption we obtain Therefore, it follows from the 7th assumption of Theorem 14 thatwhich ensures that . On the other hand, since is bounded in and is ball compact then there exists a subsequence which converges to some limit . By Algorithm 13 this subsequence satisfiesThus, by letting in the inequality (58) and by taking into account the upper semicontinuity of and the continuity of and , we obtain This means that is a solution of . Finally, using now Proposition 12 we get is a solution of and so the proof is complete.

4.2. Case 2: Is a General Set-Valued Mapping

In this general case we propose the following algorithm.

Algorithm 15. Let and for all ; (1)Select ;(2)For select such that  where is a given positive number and is the image of , that is, .Obviously Algorithm 15 coincides with Algorithm 13 when is a constant set-valued mapping. However the assumptions assumed on in the previous subsection are not sufficient to prove the convergence of the sequence generated by Algorithm 15 to a solution of . We need to replace the -strong monotonicity by a relaxed -strong monotonicity of the bifunction over and we do not assume the nonemptiness of the solution set of the proposed problem. We will say that is relaxed -strongly monotone over provided that for some we haveBy symmetry of , it is clear that any -relaxed strongly monotone bifunction with respect to is -strongly monotone with respect to . This relaxed assumption on has been used in Hilbert spaces in [4] and in Banach spaces in [13]. The following theorem is our main result in this subsection.

Theorem 16. Let be a -uniformly convex Banach space. Let be a closed nonempty subset of and let be a bifunction satisfying for any . Let be a sequence generated by Algorithm 15. Assume that (1)The values of are -uniformly prox-regular with some ratio ;(2)The image of is ball compact in and its graph is closed;(3) is relaxed -strongly monotone over with some ;(4) is upper semicontinuous with respect to the first variable over ;(5) is -uniformly prox-regular over with some ;(6)There exists such that for all ;(7)The nonnegative parameter is taken in the interval .Then, there exists subsequence of converging to a solution of .

Proof. Let . By the relaxed -strong monotonicity of over we have By Algorithm 15 we have with . Combining these two inequalities we getTherefore,Define now the same nonnegative real sequence used in the proof of Theorem 14. Then we have which ensures with (65) that Using the assumption yields Following the same reasoning in the proof of Theorem 14 and the ball compactness of the image of , we get a subsequence which converges to some limit satisfying by closedness of the graph of . By Algorithm 15 this subsequence satisfiesThus, by letting in the inequality (69) and by taking into account the upper semicontinuity of and the continuity of and , we obtain This means that is a solution of which ensures by Proposition 12 that under the assumptions of our theorem the solution is also a solution of . Thus completing the proof.

4.3. Case 3: Has the Form:

In this subsection we restrict our attention to the following form of the bifunction : where is a nonlinear operator. In this case and so becomes: We suggest the following algorithm to solve under some natural and appropriate assumptions on and .

Algorithm 17. Let with be too small. (i) Select and ;(ii) For ,(a) Compute ;(b) Compute and , where is the generalised projection defined in terms of the functional instead of the norm square (introduced and studied in the convex case in [16] and for the nonconvex case we refer to the recent paper [11]). A point is called the generalised projection of a given provided that The following characterisation of the -proximal normal cone in terms of the generalised projection is proved in [1].

Proposition 18. For any closed nonempty set in a reflexive Banach space and for any point we have

We need the following lemma:

Lemma 19. Let be a closed set in , , and . If  ; then , for any .

Proof. Let , and let be a point satisfying . Assume that . Let . We claim that First, observe that for any we have If , then obviously we haveOtherwise, we have . Then since we have and so we obtain that is,Therefore, from (76) and (79) we have in both cases Hence On the other hand we have the decomposition Consequently, we have that is, which means that and hence the proof is complete.

Now, we state and prove our main theorem for .

Theorem 20. Let be a -uniformly smooth Banach space. Let be a set-valued mapping with closed nonempty values and . Let be a sequence generated by Algorithm 17. Assume that(1)The solution set of is nonempty;(2) is bounded by some constant ;(3) is -Lipschitz, with constant ; that is, (4) is -strongly monotone with constant ; that is, (5)The values of satisfy for some :  for any unit vector in and any solution of ;(6)There exists some constant and such that for all ;(7)The positive constants and satisfy the inequality ;(8)The parameter in Algorithm 17 satisfies Then, the sequence generated by Algorithm 17 converges to a solution of .

Proof. Let be a solution of , that is, . Then by the characterisation of the -proximal normal cone in Proposition 18, there exists such that . Using Lemma 19 we obtain , for any . By assumption (5) we may assume that and so we get . Hence for . Since is 2-uniformly smooth we have is 2-uniformly convex; that is, for some constant (depending only on the space ) and so by Lemma 7 we get Thus we can write Therefore, Using the -Lipschitz continuity of with ratio we have and by the -strong monotonicity of with ratio we have Thus, we get and so On the other hand we have by the 6th assumption ThusOur assumptions and the choice of ensure that and hence which means that by the uniform continuity of and thus completing the proof.

Remark 21. A simple inspection of the proof of the previous theorem shows that the result is valid in the case when is taken a general set-valued mapping instead of a single-valued operator defined from to and of course the assumptions on should be adapted naturally for the set-valued case.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This research was supported for the first author by the NSTIP Strategic Technologies Program in the Kingdom of Saudi Arabia, Award no. (11-MAT1916-02).