Hybrid Alternated Inertial Projection and Contraction Algorithm for Solving Bilevel Variational Inequality Problems

Abuchu, Jacob Ashiwere; Ofem, Austine Efut; Ugwunnadi, Godwin Chidi; Narain, Ojen Kumar; Hussain, Azhar

doi:https://doi.org/10.1155/2023/3185746

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 3185746 | https://doi.org/10.1155/2023/3185746

Hybrid Alternated Inertial Projection and Contraction Algorithm for Solving Bilevel Variational Inequality Problems

Jacob Ashiwere Abuchu,^1,2Austine Efut Ofem,¹Godwin Chidi Ugwunnadi,^3,4Ojen Kumar Narain,¹and Azhar Hussain⁵

Academic Editor: Xian-Ming Gu

Received20 Oct 2022

Revised04 Jan 2023

Accepted25 Jan 2023

Published22 Mar 2023

Abstract

This paper considers an alternated inertial-type extrapolation algorithm for solving bilevel pseudomonotone variational inequality problem in the framework of real Hilbert spaces with split variational inequality and fixed-point constraints of demimetric mapping. The algorithm which involves alternated inertial uses self-adjustment stepsize condition that depends solely on the information from previous iterative step. The bilevel problem considered in the work consists of upper-level problem with an underlying operator which is pseudomonotone, while the lower problem is associated with strongly monotone mapping and the fixed-point constraint of demimetric mapping. Our algorithm is anchored on modified projection and contraction techniques, fused with alternated inertial and relaxation. Under some suitable conditions on the algorithm control parameters, we obtain a strong convergence result of the proposed method without the prior knowledge of the operator norm or the coefficients of the underlying operators within the scope of real Hilbert spaces. Finally, some numerical examples are presented to illustrate the gain of our method in comparison to some related algorithms in the literature.

1. Introduction

Let be a nonempty, closed and convex subset of a real Hilbert space that is embedded with the inner product , and associated norm . Let and be real single-valued nonlinear mappings. Then, bilevel variational inequality problem (BVIP) introduced by Anh et al. [1] is formulated as follows:where is the solution set of the variational inequality problem (VIP):

It is obvious from (2) that bilevel variational inequality problem is a classical problem containing another VIP as constraints. The variational inequality (2) is a very efficient tool in nonlinear analysis and mathematics in general. The concept of VIP can be first traced to Stampacchia [2] (it was also independently formulated by Fichera [3]) as a tool for modeling dynamical problems in mechanics. It provides a unified structure for investigating many problems arising in pure and applied sciences, structural analysis, and other areas (see [4–6] and references therein). Due to the importance associated with the problem (2), it has been extensively studied and extended in many directions. It is very interesting to note that finding the solution of VIP (2) is equivalent to solving the fixed-point problem (FPP):where is strictly positive real number and is metric projection of onto . It can be seen in [7], [Section 4, p. 314] that solves if and only if solves (3).

In 2012, Censor et al. [8] studied and extended the notion of VIP (2) to split variational inequality problem (SVIP) which entails:where and are nonempty closed and convex subsets of the Hilbert spaces and , , are nonlinear mappings and is a bounded linear operator.

From the knowledge of (3) emerged another interesting extension and generalization of VIP (2) known as split variational inequality and fixed-point problem (SVIFPP) introduced by Hai et al. in [9]. This notion involveswhere and are nonempty closed and convex subsets of the Hilbert spaces and , respectively, , are nonlinear mappings and is bounded linear operator. If in (4) and (5) we let , we obtain a special form of SVIP known as split feasibility problem (SFP):

Similarly, if in (6) and (7) we let and (an identity operator), the split variational inequality and fixed-point problem reduces to finding the common elements of the variational inequality and fixed-point problems. That is

From (9) and (10) we can deduce that , where is the fixed point set of the operator defined by .

In 2008, Mainge introduced and studied another special form of bilevel variational inequality problem :

He [10] introduced an iterative algorithm for approximating common solution of the bilevel variational inequality problem and fixed-point problem (11). The algorithm entailswhere is monotone and - Lipschitzian, is strongly monotone and Lipschitz continuous, is -demicontractive, is a metric projection of onto . Under certain conditions on the algorithm parameters, the author established a strong convergence result. But one noticeable defect of the algorithm is the double calculation of the projection onto the feasible set per iteration which can be computationally very expensive if is structurally complex.

Inspired by the work of Mainge, the authors in [9] introduced and analyzed iterative algorithm for solving another special form of (1) as given in the following:where , is strongly monotone and Lipschitz continuous on , is pseudomonotone and Lipschitzian on , is bounded linear operator and is demicontraction and demiclosed at the origin. Their iterative algorithm for solving (13) was based on subgradient extragradient technique with weaker condition on the cost operator compared to the one in [11]. They established a minimum-norm convergence result based on certain assumptions on the algorithm control parameters. Conspicuous among is the strict assumption that selection of stepsize depends on the operator norm which is not generally very easy to compute in most instances. This in turn will affect the efficiency of the iterative scheme. Stepsizes, as we know, play vital roles in the convergence properties of iterative algorithms, since the efficiency of iterative methods essentially depends on it and the knowledge of the operator norm or coefficient of the operator and otherwise.

Recently, motivated by the work in [9], the authors in [12] introduced and studied an inertial iterative algorithm for solving BVIP (13) in real Hilbert spaces. The algorithm entailswhere is a half-space constructed as: , is pseudomonotone and Lipschitzian on , is strongly monotone and Lipschitz continuous on , is a bounded linear operator, is an identity operator, , is demicontractive mapping resulting from the composition of demicontractive mapping and directed mapping such that is restricted to set of points in , the stepsizes and are self-adaptive following some given rules. The inertial parameter satisfies the condition such that with . Under these control parameters, the authors obtained a strong convergence result.

In investigating the evolving trend of research in solving the bilevel variational inequality (1) and the related problems (for instance (2), (4), (5), (11), (13)), many techniques and methods have been developed. These include gradient projection method [13], extragradient method [14, 15], subgradient extragradient method [9, 16, 17], and so on. It is worth mentioning here that these methods come with their associated defects ranging from stringent conditions on the cost operators, only weak convergence, double calculation of metric projections onto a feasible set per iteration (which is not computationally cost effective), strict conditions on algorithm control parameters like prior knowledge of Lipschitz constant, computation of coefficients of operators and so on. To effectively handle some of these shortcomings, the projection and contraction method (PCM) was introduced for solving variational inequalities and other related problems. He in [10] introduced and analyzed the following PCM algorithm:where , and , is monotone and Lipschitz continuous and is always updated by some certain self-adaptive rules. It was established that the sequence generated by (15) converges weakly to the solution of VIP (2). The projection and contraction method has enjoyed several modifications and generalizations (see [18–20] and references therein).

Recently, Viet Thong et al. in [21] introduced the following iterative algorithm for solving BVIP (1):where is -Lipschitzian, pseudomonotone, and sequentially weakly continuous, is -strongly monotone and -Lipschitz continuous, , , such that and . The adaptive stepsize is updated as follows:

The authors established that under some appropriate assumptions, the sequence generated by (16) converged strongly to unique solution of BVIP (1).

Also, Tan et al. [22] introduced and analyzed an accelerated projection and contraction extragradient algorithm for solving bilevel variational inequality problem (1). They proposed the iterative method:where is -Lipschitzian, pseudomonotone, and sequentially weakly continuous, is -strongly monotone and -Lipschitz continuous, , , such that and . The stepsize , is updated as follows:

The authors under appropriate assumption on the algorithm parameters established that algorithm (18) converges strongly to the unique solution of BVIP (1). The PCM technique has enjoyed modifications in several ways. Inertial-type of projection and contraction algorithms for solving variational inequality problems is one of the most interesting and important extrapolation techniques which is geared toward improving the rate of convergence of the iterative schemes (see for example [23–25] and references therein). This is because the introduction of inertial into iterative algorithm has been shown to efficiently accelerate the rate of convergence of such iterative method.

However, it has been observed that there are some attractive properties of projection and contraction method that do not hold in inertial projection method. For instance, the inertial term looses its Fejr monotonicity for the real sequence for each (see for example [18, 25, 26]). This ensures that the sequence generated swings back and forth within the solution set . This has a negative effect on the rate of convergence of the inertial iterative algorithm comparatively to the noninertial ones. To mitigate this problem, Mu and Peng [27] introduced and studied alternated inertial methods which applies inertial every other iteration. Since then, variants of alternated inertial algorithms have been developed to solve variational inequalities and other related problems (check e. g [28–30]). The alternated inertial term is generated by the iterative process:

This form of inertial term (20) is quite unpopular compare to the usual vanilla inertial extrapolation step found in (14) (see also the inertial terms in [18, 31]). However, the alternated inertia has very good convergence properties and performance.

Motivated by these results, we introduce and study a modified, relaxed, and alternated inertial extrapolation algorithm for finding the minimum-norm solution of bilevel variational inequality problem of the type:wherewhere , are real Hilbert spaces, is strongly monotone and Lipschitz continuous, is pseudomonotone and Lipschitzian, is bounded linear operator, is demimetric and demiclosed at the origin. Our proposed iterative method is based on alternated inertial extrapolation and relaxation step under adaptive stepsize condition in real Hilbert spaces. Under some mild assumptions on the algorithm control parameters, we establish a strong convergence result without the prior knowledge of the operator norm or the coefficients of the underlying operators. We also present some numerical illustrations to show the applicability and efficiency of our method.

2. Preliminaries

In this section, we will present some definitions and results that will be useful in our main results.

Let be a real Hilbert space and a nonempty, closed and convex subset of . For every there exists a unique nearest point in denoted such that

The operator is known as the metric projection of onto and is known to be nonexpansive. Moreover, has the following characterization:

Lemma 1 (see [32]). Let be metric projection of onto . Then, the following hold:(i);(ii);(iii).

Definition 1 (see [33, 34]).. Let be real Hilbert space and a nonempty, closed and convex subset of . Let be a real single-valued mapping and denotes the set of all fixed points of . Then, is said to be as(a)-Lipschitz continuous on if there exists such that(b) strongly monotone on if there exists such that(c)Monotone on if(d)Pseudomonotone on if(e) demicontractive operator on if there exists such that(f) demimetric on if there exists such thatIt is obvious to see that (b) (c) (d). But the converses are not generally true.

Definition 2 (see [30]).. A sequence is said to be Feje'r monotone with respect to a set if each point in the sequence is not strictly farther from any point in than its predecessor. That is

Lemma 2 (see [33, 35]).. Let be a real Hilbert space with and , then the following holds(i);(ii);(iii).

Lemma 3 (see [36]).. Let be a nonempty, closed and convex subset of a real Hilbert space , and be a -strongly monotone and Lipschitz continuous mapping on . Let and . Then, for any nonexpansive mapping , we can associate a mapping defined by . Then, is a contraction mapping. That iswhere .

Lemma 4 (see ([37] Lemma 3.3)).. Let be a nonempty, closed, and convex subsets of a real Hilbert space and be a pseudomonotone and Lipschitz continuous operator. Suppose is a real sequence and is a subsequence of a sequence in such that , and , where is a subsequence of the sequence , then .

Lemma 5 (see [38]).. Let be a sequence of positive real numbers. Let and be sequences in with . Suppose that satisfies the inequality:If the for every subsequence of of satisfying the condition . Then, .

3. Main Results

Throughout the remaining part of this work, we shall use (respectively, ) to denote that the sequence converges strongly (respectively weakly) to a point as . We shall also make the following assumptions that will help us establish convergence result for our proposed iterative algorithm.

Assumption 1. Supposed that:(A1) is a nonempty, closed and convex subset of the real Hilbert space ;(A2) is pseudomonotone operator and Lipschitz continuous on , ;(A3) is strongly monotone and Lipschitz continuous on , ;(A4) is demimetric mapping and demiclosed at the origin, ;(A5) is a nonzero bounded linear operator with adjoint ;(A6) is a real sequence defined in and satisfying the property: , a positive real sequence with , i.e., and , , ;(A7)The solution set of BVIP (21), .

	Initialization: Choose , (], , , and given the initial points arbitrarily.
	Iterative Process Steps: Given the iterates and , choose such that , where

	We calculate the iterate as follows:
	Step 1: Set , we compute the iterate as follows:

	and compute

	where the stepsize is updated as

	If , stop. is the required solution. Else, execute Step 2.
	Step 2: Compute

	where

	Step 3: Compute

	where for any fixed , is chosen such that

	otherwise set .
	Set and return to Step 1.

Remark 1. It is pertinent we give some of the following highlights of Algorithm 1 in comparison to some methods in the literature.(1)In [10–12], algorithms for solving two level variational inequality problems in the framework of real Hilbert spaces with a fixed-point constraint of demicontractive mapping were introduced by the authors. Their algorithms were based on the extragradient method and subgradient extragradient method with underlying class of monotone and inverse strongly monotone operators on respectively. In this work we proposed a modified projection and contraction algorithm that is based on alternated inertial extrapolation and relaxation for solving bilevel variational inequality problem with a fixed-point constraint of demimetric mapping in the setting of real Hilbert spaces when the underlying operators are pseudomonotone and strongly monotone. It is worth mentioning here that the class of operator associated with the upper problem (pseudomonotone operators) and fixed-point constraint (demimetric mapping) considered in this work is more general than the ones in [10–12] and include several important classes of operators. Specifically, pseudomonotone operators enjoy the inclusion of strongly monotone operators, monotone operators, and several others. Also, a class of demimetric mappings includes nonexpansive mappings, demicontractive mappings, directed mappings, and others.(2)The proposed method involves alternated inertia term as well as relaxation which are very key in improving the rate of convergence for solving BVIP (21). Notably, the alternated inertia unlike the usual vanilla inertia plays integral role in the recovery of monotonicity status of the sequence for each in the solution set . This helps in improving the rate of convergence of the iterative scheme.(3)In our convergence analysis, the proof of the main result; Theorem 1 (strong convergence) does not follow the traditional way of “Two Cases Approach” found in many articles handling optimization problems (see [33, 39, 40]).(4)In our proposed Algorithm 1, the predictive stepsize conditions and use self-adaptive methods which are very simple and do not require the priori knowledge of the operator norm unlike algorithm in [37] whose stepsize depends on operator norm which may not be easy to compute.

4. Convergence Analysis

Remark 2. Note from Algorithm 1 Step 1, it can be observe that . Thus, for , then we haveThis implies that . Since the sequence is monotone decreasing, the limit exists. Thus, there exists such that .

Lemma 6. Suppose is a sequence generated by Algorithm 1 under Assumption 1. Then, is bounded

Proof. Let . Then, if is odd, since , we haveSince the operator is pseudomonotone, we haveSince the stepsize , it follows thatBy Lemma 1, we haveBy adding (36) and (37) we obtainFrom Algorithm 1, Step 2 and (38), we getBut we observe thatSince is a projection onto , it follows thatThus, from (39), (40), and (41) we getBut we observe from the definition of the sequence in Algorithm 1 Step 2 that for each ,From the definition of , we know thatIt follows thatCombining (43) and (45) we obtainFrom Algorithm 1 Step 2 we haveImplying thatThus,From (46) and (49) we getUsing the definition of the sequence in Algorithm 1 Step 3, Lemma 2 we getSince the operator is demimetric, from (52) we getBut we know from the definition of the adaptive stepsize in (34) that for we haveIt follows thatThis impliesImplying,Hence,Combining (53) and (58) we obtainAlso using the definition of the sequence in Algorithm 1 Step 1 and for each we haveFrom Algorithm 1, we know that . By (A6) it follows thatThis implies that the sequence is bounded. And so, there exists a positive constant such that for all . Thus, from (61) we getUsing the definition of in Algorithm 1 (52), (60), (63), and Lemma 3 we obtainwhere .
Following similar line of argument we getHence, by inductionSince is bounded, it implies that is bounded. Consequently, , , , , and are all bounded.

Lemma 7. Let be sequence generated by Algorithm 1 under Assumption 1 such that . Then, satisfies the inequalitywherewith as defined in (A6), is a real constant and .

Proof. Let . Then, from the definition of in Step 3 of Algorithm 1 under Assumption 1, using Lemma 2, Lemma 3, (51), and (59) we obtainBut we observe thatwhere .
By combining (69) and (70) we havewhich yields the desired result.

Theorem 1. Suppose is the sequence generated by Algorithm 1 under Assumption 1. Then, converges strongly to which solves BVIP (21), where

Proof. Let . Using the definition of in Algorithm 1 under Assumption 1 and following the same technique in Lemma 7 we obtainUsing Lemma 5 and (76) we need establish thatfor each subsequence of the sequence satisfying the condition:Let be the subsequence of satisfying condition (78), thenFrom (75) we haveThis follows thatThus,Observe from the definition of the sequence thatThis follows thatAlso, we observe thatBut from Algorithm 1 Step 2 (setting since is even), we haveIt impliesCombining (85), (86), and (88) we obtainIt followsFrom Remark 2, we know thatThis shows that the sequence is bounded. Thus, by combining (82) and (90) we getSince the sequence is bounded and the operator is demimetric, there exists a real positive constant such thatBut , for any since is bounded. Therefore, we have,Consequently,Combining (83) and (95) we getSimilarly, using (82) and (92)Thus,Using (83) we getFrom (82) and (99) we obtainSince as , using (100) and condition (A6) we haveSince is bounded and , there exists subsequence of such that as . From Lemma 4 we obtain that .
Also, since , from (82) we have as . We know that the operator is bounded and so, we have that converges weakly to . Thus, combining (110) and demiclosedness of the operator , we get that .
Lastly, we need establish that as which solves BVIP (21). But we know that converge weakly to , henceSince is a unique solution of (21), then and it follows thatThus, using (101) and (103) we obtain,Applying Lemma 7, we obtainTherefore, as establishing strong convergence result for sequence of even terms.
Next we establish convergence for sequence of odd terms . From (71) and (101) and the condition on the control sequences in (A6), we getHence,Following the same argument in obtaining (82), (83), and (96) we get,Also, following similar approach of getting (98), (99), and (100) we obtainUsing (111) and Assumption (A6) we getThus, following the same argument used even case, we show thatAnd so by Lemma 5 we obtainTherefore, the sequence converges strongly to completing the proof.

5. Numerical Examples

Under this section, we shall present numerical illustrations to show the behavior of our proposed iterative method, Algorithm 1 in comparison to algorithms of Maing (12), Akutsah et al. (14), and Thong et al. (16). All codes are written in Matlab R2018a and executed on a PC Desktop Intel (R) Core (TM) i5-6300U CPU @ 2.40 GHz 2.50 GHz, RAM 8.00 GB. In the numerical results presented in the table below, “No of Iter.” denotes the number of iterations, and “Sec.” denotes the running time of CPU in seconds. Also, for the purpose of these numerical implementations, we shall choose , , , , , , , , , and for all . It is obvious that, these parameters satisfy the conditions in Assumption 1. Similarly, we choose for algorithm of Maing, for algorithm of Akutsah et al., and for algorithm of Thong et al.

Example 1. Let be a linear spaces of square summable sequences. That iswith the associated norm and inner product where and . Let and be mappings defined by with and with , respectively. Consider a mapping such that . as defined is a bounded linear operator with adjoint . Let the feasible set be defined by . Finally, let be a mapping defined by . We shall take as the stopping criterion and choose the following as starting points: Case A: and . Case B: and . Case C: and . Case D: and .

Example 2. Let be linear spaces of square integrable functions embedded with inner product and associated norm . Consider the feasible . Define a projection from onto byConsider operator given by and Volterra integral operator defined by and . is linear, monotone and Lipschitzian (see [41]). Let be a mapping defined by . Given that , it implies thatThis shows that is pseudomonotone. Define by . is demimetric. Also let be a bounded linear mapping defined by with adjoint and . Taking as the stopping criterion we obtain the following results using the following as starting points: Case A: and . Case B: and . Case C: and . Case D: and .

Example 3. Let and . Consider a mapping defined by , where is an matrix and is a column vector in . It can be easily seen that is monotone and Lipschitzian with Lipschitz constant . Let be a mapping defined by , where is symmetric matrix which is positive definite and is vector in . defined above is strongly monotone and Lipschitz continuous. Let . Consider the bounded linear operator generated randomly in the interval (0, 1) and mapping define by . We shall consider different cases of dimensions , and (250, 25) with and randomly generated in . Also we take as the stopping criterion. Using the appropriate parameters we obtain the following results:
With the examples (Example 1–Example 3, and cases (Case A–Case D) for each, we compare Algorithm 1 with Algorithm (12) of Mainge in [11], Algorithm (16) of Thong et al. in [21] and Algorithm (14) of Akutsah et al. in [12], Tables 1–3 and Figures 1–3 show that our algorithm performs better than the Algorithms in Mainge, Thong et al. and Akutsah et al. in terms of number of iterations steps and CPU running time. Also, in Example 3 we compare Algorithm 1 with Algorithm (16) of Thong et al. in [21] and Algorithm (14) of Akutsah et al. in [12] which the results validate the fact that our proposed algorithm performs better.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

6. Conclusion

We introduce and study a modified relaxed alternated inertial projection and contraction algorithm for solving bilevel split variational inequality problem when the underlying mapping is pseudomonotone in real Hilbert spaces with a fixed-point constraint of demimetric mapping. Our proposed iterative algorithm is based on alternated inertial extrapolation fused with relaxation under adaptive stepsize conditions in the framework of real Hilbert spaces. Under some certain assumptions on the algorithm parameters, we establish that the sequence generated by the algorithm converges strongly to a unique solution of bilevel pseudomonotone variational inequality problem without the prior knowledge of the operator norm or the coefficients of the underlying of operators. Finally, some numerical examples were presented which validate the applicability and efficiency of our iterative method.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

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Copyright © 2023 Jacob Ashiwere Abuchu et al. 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.

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