# A Self-Adaptive Extragradient Algorithm for Solving Quasimonotone Variational Inequalities

Authors: Li-Jun Zhu; Tzu-Chien Yin
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This article aims to research iterative schemes for searching a solution of a quasimonotone variational inequality in a Hilbert space. For solving this quasimonotone variational inequality, we propose an iterative procedure which combines a self-adaptive rule and the extragradient algorithm. We demonstrate that the procedure weakly converges to the solution of the investigated quasimonotone variational inequality provided the considered operator satisfies several additional conditions.


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## Body

## 1. Introduction

Variational inequality emerged in 1964 arising from the study of mechanics has many applications in engineering, economics, operations research, etc. ([1–4]). Variational inequality theory acts as a tool for solving many problems, such as equilibrium problems ([5, 6]), optimization problems ([7–9]), fixed point problems ([10–12]), and split problems ([13–17]). There are numerous iterative schemes for solving variational inequalities in the existing results; see [18–25]. Next, we briefly review several valuable iterative methods.Throughout, suppose thatH is a Hilbert space. The symbols ⋅,⋅ and ⋅ are the inner and norm of H, respectively. Let ∅≠D⊂H be a convex and closed set. For an operator ϕ:D⟶H, the variational inequality aims to seek a point v∈D satisfying
(1)ϕv,v−v^≤0,∀v^∈D.We useSϕ,D to indicate the set of solutions of (1).A valuable algorithm for solving (1) is the projection algorithm ([26, 27]) which generates a procedure as follows:
(2)s0∈D,sn+1=projDsn−ρϕsn,foralln≥0,where projD:H⟶D stands for the orthogonal projection and ρ>0 means the step-size.Whenϕ is strongly monotone, strongly pseudomonotone, or inverse strongly monotone, iterative scheme (2) is convergent ([28, 29]).Another powerful method is extragradient method studied by Korpelevich [30] which generates a procedure starting from an initial point s0∈D:
(3)tn=projDsn−ρnϕsn,sn+1=projDsn−ρnϕtn,foralln≥0.Thereafter, (3) has been discussed extensively for solving (1); see, e.g., [30–34]. The main reason why the extragradient method attracts so much attention is that extragradient method can be used to find a solution of plain monotone operators. In fact, extragradient algorithm can be used to solve (1) if ϕ is pseudomonotone and sequentially weakly continuous ([35–37]).Very recently, iterative methods for solving quasimonotone variational inequality have been investigated in the literature [24, 38, 39]. Especially, Salahuddin [40] utilized (3) for solving a Lipschitz quasimonotone variational inequality and achieve the following result.Conclusion 1 ([40]).
Assume that the operatorϕ satisfies (i) quasimonotone on H; (ii) sequentially weakly continuous on D; and (iii) Lipschitz continuous on D. Suppose that Sϕ,D≠∅ and ρn∈a,b⊂0,1/L,∀n≥0. Then, sn obtained from (3) weakly converges to z^∈Sϕ,D.In this article, we further utilize extragradient method (3) for solving quasimonotone variational inequality (1). For this task, we will make use of an auxiliary tool regarding the following dual variational inequality which is to find v∈D satisfying:
(4)ϕv^,v−v^≤0,∀v^∈D.We useSdϕ,D to indicate the set of solutions of (4).Notice thatSdϕ,D is closed convex. At the same time, we have Sdϕ,D⊂Sϕ,D when ϕ is continuous and D is convex. However, to acquire the convergence of the constructed sequence, one has to add the following extra condition
(5)Sϕ,D⊂Sdϕ,D,which implies that
(6)ϕz,z−z†≥0,∀z†∈Sϕ,Dandz∈D.Note that the above condition (5) holds if ϕ is pseudomonotone. However, this condition (5) is not satisfied when ϕ is quasimonotone. Further, self-adaptive rule was applied for solving variational inequality problems, see [41–45]. In this paper, for solving quasimonotone variational inequality (1), we propose an iterative procedure which combines a self-adaptive method and extragradient method (3) without using condition (5). We show that the suggested iterative procedure is weakly convergent. Our result extends the above theorem (1) at two aspects: on the one hand “sequential weak continuity” imposed on ϕ can be replaced by a more general restriction and on the other hand a self-adaptive technique is used to relax Lipschitz condition of ϕ.


## 2. Notions and Lemmas

Throughout, suppose thatH is a Hilbert space and ∅≠D⊂H is convex and closed. A map ϕ:D⟶H is called
(1)
Monotone if(7)ϕq−ϕq^,q−q^≥0,∀q,q^∈D.(2)
Pseudomonotone if(8)ϕq^,q−q^≥0⟹ϕq,q−q^≥0,∀q,q^∈D.(3)
Quasimonotone if(9)ϕq^,q−v^>0⟹ϕq,q−q^≥0,∀q,q^∈D.By the above definition, we can deduce that ifϕ is pseudomonotone, then ϕ must be quasimonotone. However, the reverse conclusion may fail.A mapϕ:D⟶H is called Lipschitz continuous if
(10)ϕq−ϕq^≤τq−q^,∀q,q^∈D,where τ is some positive constant. In this case, we call ϕτ-Lipschitz. ϕ is called nonexpansive provided τ=1.An orthogonal projection fromH onto D, denoted by projD fulfills
(11)v∈H,v−projDv≤v^−v,∀v^∈D.projD possesses the following characteristic inequality:
(12)v∈H,v−projDv,v^−projDv≤0,∀v^∈D.


## 3. Algorithms and Convergence Results

First, we declare several related conditions. Suppose thatH is a Hilbert space and ∅≠D⊂H is convex and closed. Suppose that the involved operator ϕ satisfies three restrictions:(t1)ϕ:H⟶H is a quasimonotone operator(t2)ϕ is μ-Lipschitz on D(t3) Iflim¯n⟶+∞ϕsn=0 with sn being a sequence in H and sn⇀s‡, then ϕs‡=0In the sequel, assume thatSdϕ,D≠∅ and the set t†∈D:ϕt†=0\Sdϕ,D is finite.Suppose thatϖ,ζ and ς^ are three constants in the open interval 0,1. Suppose that τn is a sequence in 0,2 satisfying 0<lim¯n⟶∞τn≤lim¯n⟶∞τn<2.Next, we state our scheme for solving (1).Algorithm 2.
Select a fixed pointu0 in D. Set n=0.
Step 1. Assumeun is presented. Compute
(13)wn=projDun−ϖζnϕun,where ζn=max1,ζ,ζ2,⋯ fulfills
(14)ζnϕwn−ϕun≤1−ς^ϖwn−un.
Step 2. Computew^n=projDun−ϕun. (2a) If w^n=un, then stop. (2b) If w^n≠un, then calculate
(15)vn=v^nv∧n2wn−un2,where v^n=un−wn+ϖζnϕwn and compute
(16)un+1=projDun−ς^τnvn.
Letn≔n+1 and return to Step 1.Conclusion 3.
Inequality (14) is well-defined. Moreover, 0<1−ς^ζ/ϖμ<ζn≤1−ς^/ϖμ or ζn=1.Proof.
Sinceϕ is μ-Lipschitz, we obtain
(17)ϕprojDun−ϖζnϕun−ϕun≤μprojDun−ϖζnϕun−un,which equals to
(18)1−ς^μϖϕprojDun−ϖζnϕun−ϕun≤1−ς^ϖprojDun−ϖζnϕun−un.
This implies that (14) holds for all ζn≤1−ς^/μϖ.
It is obviously that(19)ϖζnζμwn−un≥ϖζnζϕwn−ϕun>1−ς^wn−un,which implies that ζn>1−ς^ζ/ϖμ>0.Conclusion 4.
i If w^n=un, then un∈Sϕ,D. (ii) If w^n≠un, then v^n>0 and (15) is well-defined.Proof.
(i)
Ifw^n=un, that is un=projDun−ϕun, by virtue of (12), we acquire(20)un−un−ϕun,v−un≥0,∀v∈D,which results in that un∈Sϕ,D.
(ii)
Takex∗∈Sdϕ,D. Notice that(21)v^n,un−x∗=un−wn+ϖζnϕwn,un−x∗=un−wn−ϖζnϕun,un−x∗+ϖζnϕun,un−x∗+ϖζnϕwn,un−wn+ϖζnϕwn,wn−x∗.
As a result ofx∗∈Sdϕ,D and un∈D, we have
(22)ϕun,un−x∗≥0.
As the same as (22), we obtain
(23)ϕwn,wn−x∗≥0,due to wn∈D.
In view of (21)-(23), we receive
(24)v^n,un−x∗≥un−wn−ϖζnϕun,un−x∗−ϖζnϕwn,wn−un=wn−un+ϖζnϕun−ϕwn,wn−un+wn−un+ϖζnϕun,x∗−wn=wn−un2−ϖζnϕwn−ϕun,wn−un+wn−un+ϖζnϕun,x∗−wn.
Owing toζn>1−ς^ζ/ϖμ>0, from (14), we get
(25)ϕwn−ϕun,wn−un≤ϕwn−ϕunwn−un≤1−ς^ϖζnun−wn2.
Using (12) of projD and (14), we acquire
(26)un−wn−ϖζnϕun,wn−x∗≥0.
In the light of (24), (25) and (26), we achieve
(27)v^n,un−x∗≥ς^wn−un2.
Ifw^n≠un, then wn≠un. Otherwise, by virtue of (12), un−un−ϖζnϕun,v−un≥0,∀v∈D which results in that un∈Sϕ,D and hence w^n=un. This leads to a contradiction. So, wn−un>0. It follows from (27) that v^n,un−x∗>0 which yields that v^n>0. Therefore, (15) is well-defined.
In this position, we prove a main theorem.Theorem 5.
un defined by Algorithm 2 weakly converges to an element in Sϕ,D.Proof.
Letx∗∈Sdϕ,D. Since projD is nonexpansive and x∗∈D, from (16) and (21), we have
(28)un+1−x∗2=projDun−ς∧τnvn−projDx∗2≤un−x∗−ς∧τnvn2=ς∧τn2vn2−2ς^τnvn,un−x∗+un−x∗2=un−x∗2−2ς^τnwn−un2v∧n2v^n,un−x∗+ς∧τn2wn−un4v∧n2.
Substituting (27) into (28) to derive
(29)un+1−x∗2≤un−x∗2−ς∧22−τnτnwn−un4v∧n2.
Noting that0<lim¯n⟶∞τn≤lim¯n⟶∞τn<2, by (29), we acquire
(30)un+1−x∗≤un−x∗,which leads to that limn⟶∞un−x∗ exists. Then, un is bounded and so is ϕun. From (13), we have
(31)wn−un=projDun−ϖζnϕun−projDun≤ϖζnϕun.
Hence,wn and ϕwn are bounded.Taking into account (29), we gain
(32)ς∧22−τnτnwn−un4v∧n2≤−un+1−x∗2+un−x∗2.This leads to(33)wn−un2v^n⟶0.Thanks to the boundedness ofv^n=un−wn+ϖζnϕwn, by (33), we have
(34)limn⟶∞wn−un=0.With the help of the Lipschitz continuity ofϕ, from (34), we deduce
(35)limn⟶∞ϕwn−ϕun=0.Based on (15) and (16), we derive
(36)un+1−un=projDun−ς^τnvn−projDun≤ς^τnvn=ς^τnwn−un2v^n.In the light of (33) and (36), we achieve
(37)limn⟶∞un+1−un=0.By (12) and (13), we deduce
(38)un−ϖζnϕun−wn,z−wn≤0,∀z∈D.So,(39)1ϖζnun−wn,z−wn+ϕun,wn−un≤ϕun,z−un,∀z∈D.Observe thatun, wn, and ϕun are bounded. According to (34) and (39), we have
(40)lim¯n⟶+∞ϕun,z−un≥0,∀z∈D.Owing toun is bounded, there is uni⊂un fulfilling uni⇀b†∈D as i⟶+∞. Taking into account (40), we attain
(41)lim¯i⟶+∞ϕuni,z−uni≥0,∀z∈D.Iflim¯i⟶+∞ϕuni=0, by uni⇀b† and ϕ verifying (t1), we get that ϕb†=0. Then, b†∈Sϕ,D.Now, we assume thatlim¯i⟶+∞ϕuni>0. Then, there is an integer m>0 fulfilling ϕuni>0 for all i≥m. By virtue of (41), we attain
(42)lim¯i⟶+∞ϕuniϕuni,z−uni≥0,∀z∈D.Letεk be a real number sequence fulfilling εk>0,εk+1<εk and εk⟶0 as k⟶+∞. Based on (42), there is nik of ni fulfilling nik≥mk≥0 and
(43)ϕunikϕunik,z−unik+εk>0,∀k≥0,∀z∈D,which results in that
(44)ϕunik,z−unik+εkϕunik>0,∀k≥0,∀z∈D.Putbk=ϕunik/ϕunik2 for all k≥0. It is easily seen that ϕunik,bk=1 for all k≥0. With the help of (44), we achieve
(45)ϕunik,εkbkϕunik+z−unik>0,∀k≥0,∀z∈D.Owing to (45) and using the quasimonotonicity of ϕ, we acquire
(46)ϕz+εkbkϕunik,εkbkϕunik+z−unik≥0,∀k≥0,∀z∈D.As a result of Lipschitz continuity ofϕ and limk⟶+∞εkbkϕunik=limk⟶+∞εk=0, we deduce ϕz+εkbkϕunik⟶ϕz as k⟶+∞. In (46), letting k⟶+∞, we receive
(47)ϕz,z−b†≥0,∀z∈D.Thus,b†∈Sdϕ,D. Therefore, ωwun⊂t†∈D:ϕt†=0∪Sdϕ,D⊂Sϕ,D. Next, we prove un has no more than one weak cluster point in Sdϕ,D. Suppose that b†∈Sdϕ,D and c‡∈Sdϕ,D are two weak cluster points of un. Then, there exist two subsequences uni⊂un and unj⊂un such that uni⇀b† and unj⇀c‡.It is obviously that(48)2un,b†−c‡=un−c‡2−un−b†2+b†2−c‡2,∀n≥0.Lettingn⟶∞ on both sides of (48), we have that limn⟶+∞un,b†−c‡ exists, denoted by a^. Therefore,
(49)limi⟶+∞uni,b†−c‡=a^=limj⟶+∞unj,b†−c‡Note thatuni⇀b† and unj⇀c‡. By (45), we obtain
(50)b†,b†−c‡=a^=b†−c‡,c‡,which yields that c‡=b†. So, un has no more than one weak cluster point in Sdϕ,D. Since the set t†∈D:ϕt†=0\Sdϕ,D is finite, we deduce that un has only finite weak cluster points in Sϕ,D. Let w1,w2,⋯,wp be p unequal weak cluster points of un in Sϕ,D. Let Γ=1,2,⋯,p and
(51)α=minwr−ws/4,r,s∈Γ,r≠s.Forwr,r∈Γ, there is unir of un fulfilling unir⇀wr when i⟶+∞. Hence,
(52)limi⟶+∞unir,wr−wswr−ws=wr,wr−wswr−ws,∀s∈Γ.For∀s≠r, we have
(53)wr,wr−ws/wr−ws=wr2−ws2/2wr−ws+wr−ws/2>α+wr2−ws2/2wr−ws.Thanks to (52) and (53), there exists an integer Nir>0 satisfying for all i≥Nir,
(54)unir∈u^:u^,wr−wswr−ws>α+wr2−ws22wr−ws,s∈Γ,s≠r.Let(55)Ωr=∩s=1,s≠rpu^:u^,wr−wswr−ws>α+wr2−ws22wr−ws.Combining (54) with (55), we obtain unir∈Ωr,∀i≥maxNir,r∈Γ.Next we demonstrate that ifn is large enough, un∈∪r=1pΩr. Suppose that there is unj of un such that unj∉∪r=1pΩr. According to the boundedness of unj, there exists a subsequence of unj, without loss of generality, still denoted by unj, which converges weakly to v^. Hence unj∉Ωr for any r∈Γ. Then, there is a subsequence unjs of unj such that ∀s≥0:
(56)unjs∉u^:u^,wr−ws/wr−ws>wr2−ws2/2wr−ws+α,s∈Γ,s≠r.So,(57)v^∉u^:u^,wr−ws/wr−ws>wr2−ws2/2wr−ws+α,s∈Γ,s≠r,which results in that v^≠wr∀r∈Γ. It leads to a contradiction. Thus, there is a large enough integer N~ such that un∈∪r=1pΩr for all n≥N~.Finally, we show thatωwun is singleton in Sϕ,D. Suppose that p≥2. Taking into account (37), there is N^≥N~ fulfilling un+1−un<α when n≥N^. Hence, there is m≥N^ fulfilling um∈Ωr and um+1∈Ωs, where r,s∈Γ and p≥2, that is
(58)um∈Ωr=∩s=1,s≠rpu^:u^,wr−ws/wr−ws>α+wr2−ws2/2wr−wsum+1∈Ωs=∩r=1,r≠spu^:u^,ws−wr/ws−wr>α+ws2−wr2/2ws−wr.Thus, we have(59)um,wr−ws/wr−ws>α+wr2−ws2/2wr−ws(60)um+1,ws−wr/ws−wr>α+ws2−wr2/2ws−wr.Thanks to (59) and (60), we acquire
(61)um−um+1,wr−wswr−ws>2α.Note that(62)um+1−un<α.By virtue of (61) and (62), we receive
(63)2α<um−um+1,wr−wswr−ws≤un−um+1<α,which is impossible. Then, ωwun is singleton in Sϕ,D. So, un weakly converges to an element in Sϕ,D.Remark 6.
A mapϕ:H⟶H is called weakly sequentially continuous, if zn⇀z~⇒ϕzn⇀ϕz~, where zn is any sequence in H.To solve (1), many existing results have imposed the above “sequential weak-to-weak continuity” condition on ϕ; see, [35, 36, 40]. We can check if ϕ satisfies sequential weak continuity and then ϕ satisfies condition (t3).


## 4. Conclusions

The main purpose of this paper is to investigate iterative algorithms for solving variational inequality (1). A powerful method to solve (1) is extragradient method (3) introduced by Korpelevich [30] where the involved operator ϕ is pseudomonotone monotone. Based on the corresponding result of Salahuddin [40], we further apply extragradient method (3) to solve quasimonotone variational inequality (1).We propose an iterative algorithm (Algorithm2) which combines a self-adaptive rule and the extragradient algorithm. In general, in order to show ωun belongs to the solution set, ϕ should be sequentially weakly continuous. In this paper, we replace these conditions by a weaker condition (t3). We demonstrate that the procedure weakly converges to the solution of the investigated quasimonotone variational inequality under several additional conditions.



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