# The Tikhonov Regularization Method for Set-Valued Variational Inequalities

Authors: Yiran He
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

This paper aims to establish the Tikhonov regularization theory for set-valued
variational inequalities. For this purpose, we firstly prove a very general existence
result for set-valued variational inequalities, provided that the mapping involved
has the so-called variational inequality property and satisfies a rather weak coercivity
condition. The result on the Tikhonov regularization improves some known results
proved for single-valued mapping.


---

## Body

## 1. Introduction

This paper discusses the generalized variational inequality problem (in short,GVIP(F,K)) which is to find x∈K and x*∈F(x) such that(1.1)〈x*,y-x〉≥0,∀y∈K,
where K is a nonempty closed convex set in ℝn and F:K→2ℝn is a set-valued mapping with nonempty values. We use GVIP(F,K) and SOL(F,K) to denote the problem (1.1) and its solution set, respectively.Generalized variational inequality has been extensively studied in the literature; see [1–7] and the references therein.The Tikhonov regularization method is an important method for the ill-posed variational inequalities; see Pages 307 and  1224 in [8]. To our best knowledge, the Tikhonov regularization method has been discussed only for the case where the mapping F is single valued. This paper develops the Tikhonov regularization method for set-valued variational inequality GVIP(F,K).As a preparation, we firstly give an existence result forGVIP(F,K). It is well known if F is upper semicontinuous with nonempty compact convex values and the set K is compact and convex, then GVIP(F,K) has a solution. If K is noncompact, one usually requires some kind of coerciveness conditions for the existence of solutions to GVIP(F,K). Thus many researchers have attempted to search coerciveness condition as weak as possible; see [2, 3, 8, 9] and the references therein. In particular, [9] proved the following result.Theorem 1.1.
IfF is single valued  and continuous, and if the following condition is satisfied:(H)
There existsn∈ℕ such that for every x∈K∖Kn, there is y∈K with ∥y∥<∥x∥ satisfying 〈F(x),x-y〉≥0,then the variational inequality has a solution.Example  3.1 in [9] shows that the condition (H) is strictly weaker than many known coerciveness conditions. So far, it is not known whether Theorem 1.1 could be extended to the situation where F is a set-valued mapping. An affirmative answer is given in Corollary 3.9 of this paper, which says that if F is upper semicontinuous with nonempty compact convex values, then the coerciveness condition (A), which reduces to the condition (H) when F is single-valued, implies that GVIP(F,K) has a solution. Actually, a more general existence result is verified in Theorem 3.8 which does not require that F have any kind of continuity. Theorem 3.8 shows that if F has the so-called variational inequality property, then the condition (A) implies that GVIP(F,K) has a solution. If the mapping F is either upper semicontinuous (with nonempty compact convex values), or quasimonotone and upper hemicontinuous (with nonempty compact convex values), then it has the variational inequality property. Thus Theorem 3.8 unifies many known existence results for GVIP(F,K).The Tikhonov regularization method has been much discussed in the literature. In particular, assuming thatK is a box and F is a single-valued mapping, [10] proved that SOL(F+εI,K) is nonempty for any ɛ>0 (here I stands for the identity mapping), provided that F is a continuous P0-function and SOL(F,K) is nonempty and bounded. This result was extended by [11] to the situation where K is a closed convex set, the mapping F is single valued, and a coercivity condition is used to replace the assumption of SOL(F,K) being bounded. It should be noted that the coercivity condition assumed in [11] does not necessarily imply that SOL(F,K) is bounded. [12] further improves the result of [11] by assuming a weaker coercivity condition. All the above results on the Tikhonov regularization assume that the mapping F is single valued. The last part of this paper aims to establish the Tikhonov regularization theory for set-valued variational inequality (1.1). Theorem 4.1 improves the main result of [11, 12] by assuming a weaker coercivity condition and by allowing F to be a set-valued mapping (without monotonicity).General variational inequalities have been extensively discussed; see [13–16]. It should be interesting to discuss the Tikhonov regularization method for general variational inequality in a similar way.


## 2. Preliminaries

Unless stated otherwise, we assume thatK⊂ℝn is a nonempty closed convex set and F:K→2ℝn is a set-valued mapping with nonempty values. For r>0, Kr:={x∈K:∥x∥≤r}.Definition 2.1.
LetF:K→2ℝn be a set-valued mapping. F is said to be(i)
monotone onK if for each pair of points x,y∈K and for all x*∈F(x) and y*∈F(y), 〈y*-x*,y-x〉≥0,(ii)
maximal monotone onK if, for any u∈K, 〈ξ-x*,u-x〉≥0 for all x∈K and all x*∈F(x) implies ξ∈F(u),(iii)
quasimonotone onK if for each pair of points x,y∈K and for all x*∈F(x) and y*∈F(y), 〈x*,y-x〉>0 implies that 〈y*,y-x〉≥0,(iv)
F is said to be upper semicontinuous at x∈K if for every open set V containing F(x), there is an open set U containing x such that F(y)⊂V for all y∈K∩U; if F is upper semicontinuous at every x∈K, we say F is upper semicontinuous on K,(v)
upper hemicontinuous onK if the restriction of F to every line segment of K is upper semicontinuous.The following result is celebrated; see [17].Lemma 2.2.
LetK be a nonempty convex subset of a Hausdorff topological vector space E, and let G:K→2E be a set-valued mapping from K into E satisfying the following properties:(i)
G is a KKM mapping: for every finite subset A of K, co⁡(A)⊂⋃x∈AG(x), where co denotes the convex hull;(ii)
G(x) is closed in E for every x∈K;(iii)
G(x0) is compact in E for some x0∈K. Then ⋂x∈KG(x)≠∅.


## 3. Existence of Solutions and Coercivity Conditions

Definition 3.1.
F is said to have variational inequality property onK if for every nonempty bounded closed convex subset D of K, GVIP(F,D) has a solution.Proposition 3.2.
The following classes of mappings have the variational inequality property:(i)
every upper semicontinuous set-valued mapping with nonempty compact convex values,(ii)
every upper hemicontinuous quasimonotone set-valued mapping with nonempty compact convex values;(iii)
ifF is a single-valued continuous mapping and T is upper hemicontinuous and monotone with nonempty compact convex values, then F+T has the variational inequality property.Proof.
(i) is well known in the literature. (ii) is verified in [18]. (iii) is a consequence of the Debrunner-Flor lemma [19] and [20, Theorem 41.1]. Indeed, let D be a bounded closed convex subset of K, and let
(3.1)ND(x):={ξ∈Rn:sup⁡y∈D〈ξ,y-x〉≤0},
the normal cone of D at x∈D. By [20, Theorem 41.1], M(x):=T(x)+ND(x) is a maximal monotone mapping. By the Debrunner-Flor lemma [19], there is u∈D such that
(3.2)〈ξ+F(u),x-u〉≥0,∀x∈D,∀ξ∈M(x).
Since M is maximal monotone, -F(u)∈M(u)≡T(u)+ND(u). By the definition of ND, u∈SOL(F+T,D).Proposition3.3 below shows that Proposition 3.2(iii) can be extended to the case where F is a set-valued mapping.Proposition 3.3.
IfF:K→2ℝn is upper semicontinuous with nonempty compact convex values and T:K→2ℝn is monotone and upper hemicontinuous on K with nonempty compact convex values, then F+T has the variational inequality property on K.Proof.
LetD be a bounded closed convex subset of K. Define G,H:D→2D by
(3.3)G(y):={x∈D:sup⁡ξ∈T(x)〈ξ,y-x〉+sup⁡ζ∈F(x)〈ζ,y-x〉≥0},H(y):={x∈D:inf⁡ξ∈T(y)〈ξ,y-x〉+sup⁡ζ∈F(x)〈ζ,y-x〉≥0}.
Since T is monotone, G(y)⊂H(y). Since F is upper semicontinuous, H(y) is closed and hence compact in D for every y∈D. Now we prove that G is a KKM map. If not, there is {y1,…,yn}⊂D and x0=∑λiyi such that x0∉⋃i=1nG(yi)(3.4)0=sup⁡ξ∈T(x0)〈ξ,∑λiyi-x0〉+sup⁡ζ∈F(x0)〈ζ,∑λiyi-x0〉≤∑λi[sup⁡ξ∈T(x0)〈ξ,yi-x0〉+sup⁡ζ∈F(x0)〈ζ,yi-x0〉]<0.
This contradiction shows that G is a KKM map, so is H. By Lemma 2.2, there is x¯∈∩y∈DH(y). Fix any y∈D. Let yt:=x¯+t(y-x¯). Then for small t∈(0,1), yt∈D and hence
(3.5)0≤inf⁡ξ∈T(yt)〈ξ,yt-x¯〉+sup⁡ζ∈F(x¯)〈ζ,yt-x¯〉=tinf⁡ξ∈T(yt)〈ξ,y-x¯〉+tsup⁡ζ∈F(x¯)〈ζ,y-x¯〉.
Dividing t>0 on both sides, we have
(3.6)sup⁡ξ∈T(yt)〈ξ,y-x¯〉+sup⁡ζ∈F(x¯)〈ζ,y-x¯〉≥0.
Letting t→0+ yields that x¯∈G(y), as T is upper hemicontinuous. Since y∈D is arbitrary, x¯∈⋂y∈DG(y):
(3.7)sup⁡ξ∈T(x¯)sup⁡ζ∈F(x¯)〈ξ+ζ,y-x¯〉≥0,∀y∈D.
Since F(x¯) and T(x¯) are compact and convex, the Sion minimax theorem implies the existence of u¯∈F(x¯) and v¯∈T(x¯) such that
(3.8)〈u¯+v¯,y-x¯〉≥0,∀y∈D.
Thus x¯ solves GVIP(F+T,D).Before making further discussion, we need to state some coercivity conditions. The relationships of these coercivity conditions are well known in the literature; however, we provide the proof for completeness.Consider the following coercivity conditions.(A)
There existsr>0 such that for every x∈K∖Kr, there is y∈K with ∥y∥<∥x∥ satisfying inf⁡x*∈F(x)〈x*,x-y〉≥0.(B)
There existsr>0 such that for every x∈K∖Kr, there is y∈Kr satisfying inf⁡x*∈F(x)〈x*,x-y〉≥0.(C)
There existsr>0 such that for every x∈K∖Kr and every x*∈F(x), there exists some y∈Kr such that 〈x*,x-y〉>0.(D)
There existsr>0 such that for every x∈K∖Kr, there exists some y∈Kr such that sup⁡y*∈F(y)〈y*,x-y〉>0.(E)
There existsy0∈K such that the set(3.9)L(y0):={x∈K:inf⁡x*∈F(x)〈x*,x-y0〉<0}is bounded, if nonempty.Proposition 3.4.
The following statements hold.(i)
(C)⇒(B) if F is of convex values.(ii)
(D)⇒(B) if F is quasimonotone.(iii)
(E)⇒(B)⇒(A).Proof.
(i) By (C), for everyx∈K∖Kr, inf⁡x*∈F(x)sup⁡y∈Kr〈x*,x-y〉≥0. Since F(x) is convex and Kr is compact convex, the Kneser minimax theorem implies that
(3.10)sup⁡y∈Krinf⁡x*∈F(x)〈x*,x-y〉=inf⁡x*∈F(x)sup⁡y∈Kr〈x*,x-y〉≥0.
Since y↦inf⁡x*∈F(x)〈x*,x-y〉 is upper semicontinuous and since Kr is compact, there is y(x)∈Kr such that
(3.11)inf⁡x*∈F(x)〈x*,x-y(x)〉=sup⁡y∈Krinf⁡x*∈F(x)〈x*,x-y〉≥0.
This verifies (B). (ii) The implication of (D) ⇒ (B) is an immediate consequence of F being quasimonotone. (iii) (E) ⇒ (B). If L(y0)=∅, then for any x∈K, inf⁡x*∈F(x)〈x*,x-y0〉≥0; thus (B) holds. If L(y0)≠∅, then by (E), there is r>0 such that L(y0)∪{y0}⊂Kr. Thus (B) is still true. (B) ⇒ (A). Let r>0 be such that (B) holds. Then x∈K∖Kr+1, x∉Kr. By (B), there is y∈Kr such that inf⁡x*∈F(x)〈x*,x-y〉≥0. Obviously, ∥y∥≤r<r+1<∥x∥. Thus (A) is verified with r replaced by r+1.Remark 3.5.
The coercivity condition (A) is actually (C’) in [3] where it is shown that if F is quasimonotone and upper hemicontinuous with nonempty compact convex values, then (A) implies that GVIP(F,K) has a solution. However, it seems unknown whether this assertion still holds if one replaces “quasimonotone and upper hemicontinuous” by “upper semicontinuous.” An affirmative answer is given by Corollary 3.9.Remark 3.6.
The coercivity condition (B) is the condition (C) in [3]. The condition (D) appears in [21]. The condition (E) appears essentially in Corollary  3.1 in [22]; see also Proposition  2.2.3 in [8]. If F is single valued, then (E) reduces to Proposition  2.2.3(a) in [8].Remark 3.7.
Example  3.1 in [9] shows that (A) does not necessarily imply (B), even if F is single-valued and continuous.From the above discussion, (A) is the weakest coercivity condition among them. [9] proved if F is single valued and continuous, then (A) implies that GVIP(F,K) has a solution. Corollary 3.9 shows that this assertion still holds even if F is a set-valued mapping.Theorem 3.8.
LetK⊂ℝn be a nonempty closed convex set, and let F:K→2ℝn be a mapping with nonempty compact convex values. Suppose that (A) holds. If F has the variational inequality property on K, then GVIP(F,K) has a solution.Proof.
Letm>r. Since Km is bounded closed convex and F has the variational inequality property, there is xm∈Km such that
(3.12)sup⁡x*∈F(xm)〈x*,y-xm〉≥0,∀y∈Km.
(i) If ∥xm∥=m, then ∥xm∥>r, and by assumption, there is y0∈K with ∥y0∥<∥xm∥ such that
(3.13)sup⁡x*∈F(xm)〈x*,y0-xm〉≤0.
Fix any y∈K. Since ∥y0∥<∥xm∥≤m, there is t∈(0,1) such that zt:=y0+t(y-y0)∈Km. It follows that
(3.14)0≤sup⁡x*∈F(xm)〈x*,zt-xm〉≤tsup⁡x*∈F(xm)〈x*,y-xm〉+(1-t)sup⁡x*∈F(xm)〈x*,y0-xm〉≤tsup⁡x*∈F(xm)〈x*,y-xm〉.
Therefore, sup⁡x*∈F(xm)〈x*,y-xm〉≥0. Since y∈K is arbitrary, the conclusion is verified. (ii) If ∥xm∥<m, then for any y∈K, there is t∈(0,1) such that
(3.15)zt:=xm+t(y-xm)∈Km.
It follows that
(3.16)0≤sup⁡x*∈F(xm)〈x*,zt-xm〉=tsup⁡x*∈F(xm)〈x*,y-xm〉.
Since y∈K is arbitrary, xm solves GVIP(F,K).Corollary 3.9.
LetK⊂ℝn be a nonempty closed convex set, and let F:K→2ℝn be upper semicontinuous set-valued mapping with nonempty compact convex values. Suppose that (A) holds. Then GVIP(F,K) has a solution.Proof.
By Proposition3.2, F has the variational inequality property. The conclusion follows immediately from Theorem 3.8.Remark 3.10.
Assuming that the mappingF is single valued and continuous, Proposition  2.2.3 in [8] and Theorem  3.2 in [9] show that SOL(F,K) is nonempty if the conditions (E) and (A) hold, respectively. Therefore, Corollary 3.9 improves Proposition  2.2.3 in [8] and Theorem  3.2 in [9]: the mapping F is set valued instead of single valued.


## 4. The Tikhonov Regularization

Theorem 4.1.
LetK be a nonempty closed convex set in ℝn, and let F:K→2ℝn be upper semicontinuous with nonempty compact convex values. If assumption (A) holds, then for any ɛ>0,(i)
GVIP(F+εI,K) has a solution;(ii)
the set{SOL(F+tI,K):t∈(0,ɛ]} is bounded.Proof.
(i) Letr be as in assumption (A). We claim that for every x∈K∖Kr, there is y∈K with ∥y∥<∥x∥ satisfying
(4.1)inf⁡x*∈F(x)〈x*+ɛx,x-y〉≥0.
Granting this, we obtain that assumption (A) is satisfied with the mapping F replaced by F+ɛI. Since F is upper semicontinuous with nonempty compact convex values and I is continuous and monotone, Theorem 3.8 and Proposition 3.3 imply that GVIP(F+εI,K) has a solution. Now we prove the claim. Since for every x∈K∖Kr, there is y∈K with ∥y∥<∥x∥ satisfying inf⁡x*∈F(x)〈x*,x-y〉≥0, we obtain
(4.2)inf⁡x*∈F(x)〈x*+ɛx,x-y〉=inf⁡x*∈F(x)〈x*,x-y〉+ɛ〈x,x-y〉=inf⁡x*∈F(x)〈x*,x-y〉+ɛ∥x∥2-ɛ〈x,y〉≥ɛ∥x∥(∥x∥-∥y∥)≥0.
(ii) Let t∈(0,ɛ] and x(t)∈SOL(F+tI,K). Then x(t)∈Kn. If not, by assumption (A), there is y(t)∈K with ∥y(t)∥<∥x(t)∥ such that
(4.3)inf⁡x*∈F(x(t))〈x*,x(t)-y(t)〉≥0.
Since x(t)∈SOL(F+tI,K) and y(t)∈K,
(4.4)0≥inf⁡x*∈F(x(t))〈x*+tx(t),x(t)-y(t)〉=inf⁡x*∈F(x(t))〈x*,x(t)-y(t)〉+t∥x(t)∥2-t〈x(t),y(t)〉≥t∥x(t)∥2-t∥x(t)∥∥y(t)∥.
Therefore ∥x(t)∥≤∥y(t)∥, a contradiction.Remark 4.2.
Theorem4.1 improves in two ways: the mapping F is set valued instead of single valued; Theorem  1 in [12] used the coercivity condition (B) while our Theorem 4.1 uses the weaker coercivity condition (A).Theorem 4.3.
LetK be a nonempty closed convex set in ℝn, and let F:K→2ℝn be upper semicontinuous with nonempty compact convex values. Assume that there is a nonempty bounded closed convex set D⊂K such that
(4.5){x∈K:sup⁡y∈Dinf⁡x*∈F(x)〈x*,x-y〉<0}isbounded,ifnonempty.
Then for any ɛ>0, GVIP(F+εI,K) has a solution and the set {SOL(F+tJ,K):t∈(0,ɛ]} is bounded.Proof.
Letr>0 be such that
(4.6)D⋃{x∈K:sup⁡y∈Dinf⁡x*∈F(x)〈x*,x-y〉<0}⊂Kr.
Then for every x∈K∖Kr,
(4.7)sup⁡y∈Dinf⁡x*∈F(x)〈x*,x-y〉≥0.
Since D is bounded closed convex and since y↦inf⁡x*∈F(x)〈x*,x-y〉 is upper semicontinuous, there is y0∈D such that
(4.8)inf⁡x*∈F(x)〈x*,x-y0〉≥0.
Since y0∈D⊂Kr, ∥y0∥<r≤∥x∥. Therefore, assumption (A) is satisfied. Then the conclusion follows from Theorem 4.1.Remark 4.4.
IfF is a single-valued mapping, and D={x0} for some x0∈K, then Theorem 4.3 reduces to Theorem  2.1 in [11].For everyɛ>0, let Aɛ⊂ℝn, and we define(4.9)limsup⁡ɛ→0+Aɛ:={x∈Rn:∃ɛn⟶0+andxn∈Aɛnsuchthatxn⟶x}.Theorem 4.5.
LetK be a nonempty closed convex set in ℝn, and let F:K→2ℝn be upper semicontinuous with nonempty compact convex values. If assumption (A) holds, then
(4.10)∅≠limsup⁡ɛ→0+SOL(F+ɛI,K)⊂SOL(F,K).Proof.
The setlimsup⁡ɛ→0+SOL(F+ɛI,K) being nonempty follows from Theorem 4.1. Let x∈limsup⁡ɛ→0+SOL(F+ɛI,K). Then there are a sequence ɛn→0+ and xn∈SOL(F+ɛnI,K) such that xn→x. This means that for some xn*∈F(xn),
(4.11)〈xn*+ɛnxn,y-xn〉≥0,∀y∈K.
Since F is upper semicontinuous with nonempty compact values, {xn*} is compact. Without loss of generality, assume lim⁡n→∞xn*=x* for some x*∈F(x). Thus for every y∈K,
(4.12)〈xn*+ɛnxn,y-xn〉=〈xn*,y-xn〉+ɛn〈xn,y〉-ɛn∥xn∥2⟶〈x*,y-x〉asn⟶∞.
It follows from (4.11) that x∈SOL(F,K).



---

*Source: 172061-2012-04-22.xml*