Inertial self-adaptive Bregman projection method for finite family of variational inequality problems in reflexive Banach spaces

This paper considers an iterative approximation of a common solution of a finite family of variational inequailties in a real reflexive Banach space. By employing the Bregman distance and projection methods, we propose an iterative algorithm which uses a newly constructed adaptive step size to avoid a dependence on the Lipschitz constants of the families of the cost operators. The algorithm is carefully constructed so that the need to find a farthest element in any of its iterate is avoided. A strong convergence theorem was proved using the proposed method. We report some numerical experiments to illustrate the performance of the algorithm and also compare with existing methods in the literature.


Introduction
Let E be a real Banach space with dual E * and f , g denotes the duality pairing between f ∈ E and g ∈ E * . Let C be a nonempty, closed and convex subset of E and F : E → E * be a nonlinear operator, we consider the variational inequality problem (VIP) in the sense of Fichera (1963;1984) which is defined as finding a point x * ∈ C such that F x * , y − x * ≥ 0, ∀ y ∈ C. (1.1) We denote the set of solution of VIP (1.1) by V I P (C, F). The theory of VIP has been researched widely due to its applications in modelling various real life problems. For instance, it is used as a model for obstacle and contact problems in mechanics, traffic equilibrium problems and so on (see, e.g. Alber 1996). It also finds application in several fields of study such as optimization theory, nonlinear programming, science and engineering (Facchinei and Pang 2003;Hartman and Stampacchia 1966;Kinderlehrer and Stampachia 2000;Lions and Stampacchia 1967). The problem of approximating a solution of the VIP has received a lot of attention in recent published articles in the area of optimization theory (see the recent works of Hieu 2017; Jolaoso et al. 2021a, b;Oyewole et al. 2021b, a). The most popular of these methods is the Extragradient method (EGM) which was first introduced by Korpelevich (1976) for solving the saddle point problem. The EGM is known to be characterized with some drawbacks in its use for the VIP (Jolaoso et al. 2021b;Jolaoso and Aphane 2020). For instance, the EGM only converges weakly when the underlying operator is monotone. Also, the execution of the EGM depends on calculating a projection onto a feasible set twice per iteration. Another drawback of the EGM is the dependence of the method on the Lipschitz constant of the cost operator. In this direction, (Censor et al. 2011a, b) introduced the subgradient extragradient method (SEGM) where the second projection was replaced by a projection onto a carefully selected constructible halfspace. It is known that this method still preserves the weakness of the EGM through the dependence of the method on the Lipschitz condition of the cost operator. To overcome this, authors have deviced several means which includes the use of an Armijo linesearch rule (a linesearch is a technique where the outside loop is determined by some inner calculation) (see Shehu and Iyiola 2019). Another means of avoiding the dependence of the method on the Lipschitz constant is by using a self adaptive method (see Oyewole et al. 2021a and the references therein). Schematic approximation of a common solution of a system of variational inequality problems (CSVIP) has been considered in the literature. We define the CSVIP as follows: Let i = 1, 2, · · · , N , C be a nonempty, closed and convex subset of a real reflexive Banach space E and F i : C → E * be nonlinear mappings. The CSVIP consists of finding a point x * ∈ C such that F i x * , y − x * ≥ 0, ∀ y ∈ C, i = 1, 2, · · · N . (1.2) We denote the set of solution of the CSVIP (1.2) by , i.e., := {z ∈ C : F i z, y − z ≥ 0, ∀ y ∈ C, i = 1, 2, · · · N }. Note that, if N = 1 in (1.2), then the CSVIP (1.2) redues to the VIP (1.1). The motivation for studying the CSVIP stems from its applications in studying several problems in applied sciences whose constraint can be modelled as CSVIP, for instance, generalized Nash equilibrium problem and utility-based bandwidth allocation problems (see, e.g., Jolaoso et al. 2021).
For studying the approximation of solution of CSVIP in the setting of real reflexive Banach spaces, Kassay et al. (2011) first proposed a Bregman projection technique for the case when F i is a Bregman-inverse strongly monotone operator. Censor et al. (2011a) introduced the subgradient exragradient method as an improvement of the extragradient method introduced by Korpelevich (1976) and Antipin (1976) in 1970's for solving the classical VIP (1.1) in finite dimensional spaces. Their results has been generalized to infinite dimensional real Hilbert spaces by many authors, see, e.g., (Censor et al. 2011b;Khanh and Vuong 2014;Solodov and Svaiter 1999;Lin et al. 2005;Malitsky 2015). Censor et al. (2012) also proposed a hybrid method for CSVIP (1.2) which requires finding the minimum of the Lipschitz constants of the cost operators as an input parameter. This idea can be very complicated in two ways; first if F i has a complex structure (as in optimal control model) and if the feasible set is not so simple for calculating the projection onto the intersection of two constructed half-spaces. Censor et al. (2012) algorithm was modified by Hieu (2017) by using a parallel method, however the new method inherits the two drawbacks mentioned earlier. Other similar methods for solving the CSVIP can be found in Anh and Phuong (2018). Recently, Kitisak et al. (2020) proposed a modified hybrid parallel extragradient method by using an Armijo line-search technique which compute the stepsize in each iteration implicitly. It is known that the line-search idea uses an inner iteration which consume additional time and memory in its computation. Jolaoso et al. (2021b) recently proposed a new self-adaptive technique as an improvement of the previous methods for solving the CSVIP with monotone operators in real Hilbert spaces. We observe that most of the above cited results on the CSVIP used the Euclidean norm and metric projections. The Bregman distance is a key substitute and generalization of the Euclidean distance, and it is induced by a chosen convex function. It has found numerous applications in optimization theory, nonlinear analysis, inverse problems, and recently machine learning; see, for instance, (Chambolle 2004;Reem et al. 2019). In addition, the use of Bregman distance allows the consideration of a general feasible set structure for the variational inequalities. In particular, we can choose Kullback-Leibler divergence (a Bregman distance on negative entropy) and obtain an explicitly calculated projection onto a simplex. For solving the VIP, Nemirovski (2004) introduced the Nemirovski-prox method which is a variant of the EGM where the projection is understood in the sense of Bregman divergence. Gibali (2018) also proposed a Bregman projection algorithm for solving the VIP (1.1) in finite dimensional space using the extragradient technique. Moreover, other methods involving the Bregman distance for solving the VIP (1.1) can be found in Jolaoso and Aphane (2020), Jolaoso et al. (2021) and references therein. Motivated by the above results, we introduce a self adaptive algorithm for approximating a common solution of a finite family of VIPs. Using Bregman distance and projections, we obtain and prove a strong convergence theorem under a pseudomonotonicity assumption of the cost operator. In particular, the following highlights the contribution of this article to the literature: (i) We note that in the work of Jolaoso et al. (2021a), the authors studied the approximation of solution of CSVIP using metric projection. Moreover, the cost operators are monotone and weakly sequentially continuous. In this paper, using Bregman projection technique, we studied the CSVIP in the settings of reflexive Banach spaces and the cost operators are pseudomonotone with weaker continuity assumption. (ii) In Kitisak et al. (2020), the authors proposed Algorithm MPHSEM which uses a line search technique known to consume additional computational time and memory. In our algorithm, the stepsize is determined by a self-adaptive process and does not require a line searching procedure. (iii) Also in Hieu (2017), Anh and Phuong (2018), Kitisak et al. (2020), the algorithms required constructing the two sets C n and Q n then computing projection of a vector onto their intersection which can be complicated. Our method in this paper does require constructing the sets C n and Q n nor the projection onto their intersection.

Preliminaries
In this section, we give some prelimnary results and definitions that we will be used in this sequel. Throughout this paper, unless stated otherwise, let C be a nonempty closed and convex subset of a real Banach space E. We denote the weak and strong convergence of a sequence (2.1) If the limit as t → 0 + in (2.1) exists for each y, then f is said to be Gâteaux differentiable at x. In this case, the gradient of f at x is the linear function ∇ f (x), which is defined by for all y ∈ E. The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable at each x ∈ int(dom f ). When the limit as t → 0 + in (2.1) is attained uniformly for any y ∈ E with ||y|| = 1, we say that f is Fréchet differentiable at x. (b) The subdifferential set of f at a point x, denoted ∂ f (x) is given by (d) The function f is said to be Legendre if and only if the following hold: The Bregman distance (see Bregman 1967) corresponding to the function f is defined by It is well documented that the Bregman distance is not a metric in the usual sense because it fails to be symmetric and does not satisfy the triangle inequality property. However, it posses the following useful property referred to as the three points identity: for x ∈ dom f and y, z ∈ int(dom f ), we have (2. 2) The Bregman distance finds applications in solving many important optimization problems (see e.g Bauschke et al. 2001;Jolaoso et al. 2021a) and the references therein. The following are common Bregman functions with their corresponding Bregman distance functions (See Beck 2017).
The Bregman projection with respect to f of x ∈ int(dom f ) for a nonempty, closed and The Bregman projection is characterized by the following identities which are the analogues of the metric projection (see Reich and Sabach 2009): Alber 1996;Censor and Lent 1981) defined by Moreover, by the subdifferential inequality, it is easy to see (e.g Kohsaka and Takahashi 2005), that In sum, if f : E → R ∪ {+∞} is a proper, convex and lower semicontinuous function, then f * : E → R ∪ {+∞} is a proper, convex, weak lower semicontinuous function (see Phelps 1993) and thus V f is convex in second variable, that is Lemma 2.2 Naraghirad and yao (2013) Let r > 0 be a constant and let f : E → R be a continuous uniformly convex function on bounded subsets of E. Then,

Main result
In this section, we introduce a strong convergent algorithm for approximating a solution of the CSVIP and then discuss its convergence analysis. First we make the following assumptions: Assumption A (A1) For each i = 1, 2, · · · , N , the mapping F i : E → E * is pseudomonotone and uniformly continuous and L i -Lipschitz continuous. However, note that the execution of our method does not require that the Lipschitz constants be known.
a function which is Legendre, uniformly Fréchet differentiable, σ -strongly convex, and bounded on bounded subsets of E.

Remark 3.2
We observe from Algorithm 3.1, that the method is self adaptive with a monotone decreasing sequence {α k }, thus the dependence of the cost operators on the Lipschitz constants is dispensed with. The convergence of the method is made faster by the incorporation of the inertial term and finally the technique is based on the Bregman projection and distances which opens the method for more applications. To see this, it is easy to see that {α k } is monotonically non-increasing. Now, from the Lipschitz continuity of F i for all i = 1, 2, · · · , N , we have Hence, the sequence {α k } as a lower bound of min min 1≤i≤N μ L i , α 0 . Thus, there exists a limit lim Lemma 3.5 Let p ∈ and let {x k } be the sequence given by Algorithm 3.1, then the following inequality holds Proof From p ∈ and (3.3), we have From p ∈ , we have that F i p, y − p ≥ 0 for all y ∈ C. Since F i is pseudomonotone for each i, we have F i y, y − p ≥ 0 for all y ∈ C, thus F i y k,i , y k,i − p ≥ 0. Therefore, we have Using this in (3.7), we obtain Substituting (3.9) into (3.8), we get We thus have from (2.3), that Proof Let p ∈ . Since μ ∈ (0, 1) and σ > 0, we have Hence, there exists N 1 ≥ 0, such that for all k ≥ N 1 , we get 1 − μα k σ α k+1 > 0. Thus, we get from Lemma 3.5 that It follows from (3.2), that Now, we see from (3.4) and Lemma 2.2, that Finally, we have from (3.5), that This implies that {D f ( p, x k )} is bounded which implies {x k } is bounded. As a consequence of this, the sequence {w k } is bounded. We obtain from the properties of C and continuity of ∇ f , that {y k,i } is bounded. It follows easily also that the sequences {z k,i } and {v k } are bounded.
Lemma 3.7 Let {w k } and {y k,i } be the sequences given in Algorithm 3.1 such that w k − y k,i → 0 as k → ∞. Let q ∈ C be the weak limit of the subsequence {w k j } of {w k } for j ∈ N. Then, q ∈ .
Proof From the definition of y k j ,i and (2.4), we get This implies Hence, (3.13) Since w k j − y k j ,i as j → ∞, we obtain by the Fréchet differentiability of f , that Thus, we have from this, (3.13) and lim j→∞ α k j > 0, that (3.14) We consider the following possible cases to show that q ∈ . Case i: Let lim inf j→∞ F i w k j = 0. Since w k j q, we obtain by condition (A1), that F i q = 0.
Hence q ∈ . Case ii: Suppose lim inf j→∞ F i w k j > 0. Let { j } be a sequence of nonnegative real numbers decreasing to zero as j increases to ∞. For each j , let N j be the smallest integer such that where u k j is the unit vector of Since F i is pseudomontone, we obtain Next, we show that lim j→∞ j u N j = 0. To see this, since w k j q and w k j − y k j ,i → 0 as j → ∞, we obtain y N j ,i q as j → ∞. By {y k,i } ⊂ C, we obtain q ∈ C. Suppose 0 < F i z , otherwise z ∈ . By the weakly lower semicontinuity of the functional F(x) , we obtain which implies that lim j→∞ j u N j = 0.
Letting j → ∞, then the right hand side of (3.16) tends to zeo.
Hence, for all x ∈ C, we have Proof Assume p ∈ . We have from (3.5), that It follows from (3.18), that (3.20) with M 1 = sup{ϒ k : k ∈ N}. We now show that {x k } converges strongly to p. To achieve this, set a k := D f ( p, x k ), then it follows from (3.19), that We will then apply Lemma 2.4. Indeed, we only need to show that lim sup Now, let {a k j } be a subsequence of {a k } satisfies (3.21). Then by (3.20) and Assumption (B1), we obtain From Assumption (B3), we obtain (3.23) Furthermore, we see from Lemma 3.5, that where M 1 is as before. Similarly, we obtain from (3.24), that lim sup It follows from (2.3), that The following are easily obtained from previous established limits: (3.28) Using (3.28), the boundedness of ∇ f and the uniform continuity of f on bounded subsets of H , we have thus, we obtain This implies D f (z k j ,i , v k j ) → 0 as j → ∞ for each i = 1, 2 · · · , N . Hence lim j→∞ z k j ,i − v k j = 0, i = 1, 2, · · · , N . (3.30) It also follows Assumption (B1) and (3.5), that Therefore, we obtain We can now show that lim sup The weak convergences of the subsequences {w k j l }, {y k j l }, {z k j l } and {v k j l } from this and previously established limits. We therefore get from (3.25) and Lemma 3.7, that q ∈ . Also from p = x 0 , we have by the charateristics of the generalized projection, that Hence, by this and (3.32), we get which implies lim sup j→∞ ϒ k j ≤ 0. We therefore conclude by Lemma 2.4 and (3.21), that the sequence {x k } converges strongly to p.
The following are some consequences of our main theorem.
(1) Let N = 1, then we obtain a result for approximating a solution of VIP problem. Thus the results of Thong et al. (2021) are corollaries of our main result in this paper. (2) Suppose H = E is a real Hilbert space, then the result obtained in Jolaoso et al. (2021b) becomes a corollary of our main theorem.

Application to equilibrium problem
Let E be a real reflexive Banach space and C be a nonempty, closed and convex subset of E.
Let G : C × C → R be a bifunction, the Equilibrium Problem (shortly, EP) with respect to G on C is defined as: Find x * ∈ C such that G(x * , y) ≥ 0, ∀ y ∈ C. (4.1) We denote the solution set of EP (4.1) by E P(C, G). The EP was earlier referred to as Ky Fan inequality due to the contribution of Fan (1972) in the development of the field. However, it gained more attention after the revisitation by Muu and Oettli (1992) and Blum and Oettli (1994) in the 90s. Many important optimization problems such as Kakutani fixed point problems, minimax problem, Nash equilibrium problem, variational inequality problem, and minimization problem can be reformulated as (1.1). The EP has also served as an important tool in the study of wide range of obstacle, unilateral, and important problems arising in various branches of pure and applied sciences; see, for instance, Brézis et al. (1972), Lyashko and Semenov (2016). More so, several algorithms have been introduced for solving the EP (4.1) Hilbert and Banach spaces (see for instance Jolaoso 2021 and references therein).
Definition 4.1 Let C ⊆ E and G : C × C → R be a bifunction. Then G is said to be Remark 4.2 Jolaoso (2021) Every monotone bifunction on C is pseudo-monotone but the converse is not true. A mapping A : C → E * is pseudo-monotone if and only if the bifunction G(x, y) = Ax, y − x is pseudo-monotone on C.
For solving the EP, we assume that the bifunction g satisfies the following: is convex lower semicontinuous and subdifferentiable on C for every fixed where u ∈ C and λ > 0.
Then {x k } converges strongly to a point u = (x 0 ).

Numerical example
In this section, we present some numerical illustrations which demonstrate the performance of our proposed algorithm. In the first example, we consider several types of the Bregman dunction f given in Example 2.1 and compare the performance of the algorithm for each function. In particular, for x ∈ R m , it is clear that randomly generated matrices such that S i is skew-symmetric, D i is positive definite diagonal matrix and q = 0. Hence the CSVIP has a unique solution. The feasible set C is defined by for a = 2. Hence the unique solution of the CSVIP = {0}. The projection onto C is calculated explicitly. We use the following parameter for our computation: μ = 0.25, θ = 0.0125, α 0 = 0.01, β k = 1 10(k+1) , δ k,i = 8 9 i + 1 N 9 N for i = 1, . . . , N and δ k,0 = 0. The initial points x 0 , x 1 ∈ R m are generated randomly. We choose the following criterion E n = x k 2 < ε, where ε = 10 −3 and test the algorithm for the following values of m and N : We present the corresponding numerical results in terms of number of iterations and CPU time-taken for the computations in Table 1 and Figure 1.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.