# Some Existence Theorems for Nonconvex Variational Inequalities

Authors: Narin Petrot
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

By using nonsmooth analysis knowledge, we provide the conditions for existence solutions of the variational inequalities problems in nonconvex setting. We also show that the strongly monotonic assumption of the mapping may not need for the existence of solutions. Consequently, the results presented in this paper can be viewed as an improvement and refinement of some known results from the literature.


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## Body

## 1. Introduction

Variational inequalities theory, which was introduced by Stampacchia [1], provides us with a simple, natural, general, and unified framework to study a wide class of problems arising in pure and applied sciences. The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On the one hand, it reveals the fundamental facts on the qualitative aspects of the solutions to important classes of problems. On the other hand, it also enables us to develop highly efficient and powerful new numerical methods for solving, for example, obstacle, unilateral, free, moving, and complex equilibrium problems.It should be pointed out that almost all the results regarding the existence and iterative schemes for solving variational inequalities and related optimizations problems are being considered in the convexity setting; see [2–5] for examples. Moreover, all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general, when the sets are nonconvex. Notice that the convexity assumption, made by researchers, has been used for guaranteeing the well definedness of the proposed iterative algorithm which depends on the projection mapping. In fact, the convexity assumption may not require for the well definedness of the projection mapping because it may be well defined, even in the nonconvex case (e.g., when the considered set is a closed subset of a finite dimensional space or a compact subset of a Hilbert space, etc.).The main aim of this paper is intending to consider the conditions for the existence solutions of some variational inequalities problems in nonconvex setting. We will make use of some recent nonsmooth analysis techniques to overcome the difficulties that arise from the nonconvexity. Also, it is worth mentioning that we have considered when the mapping may not satisfy the strongly monotonic assumption. In this sense, our result represents an improvement and refinement of the known results.


## 2. Preliminaries

Letℋ be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let C be a nonempty closed subset of ℋ. We denote by dC(·) the usual distance function to the subset C; that is, dC(u)=infv∈C∥u-v∥. Let us recall the following well-known definitions and some auxiliary results of nonlinear convex analysis and nonsmooth analysis.Definition 2.1.
Letu∈ℋ be a point not lying in C. A point v∈C is called a closest point or a projection of u onto C if dC(u)=∥u-v∥. The set of all such closest points is denoted by projC(u); that is,
(2.1)projC(u)={v∈C:dC(u)=∥u-v∥}.Definition 2.2.
LetC be a subset of ℋ. The proximal normal cone to C at x is given by
(2.2)NCP(x)={z∈H:∃ρ>0;x∈projC(x+ρz)}.The following characterization ofNCP(x) can be found in [6].Lemma 2.3.
LetC be a closed subset of a Hilbert space ℋ. Then,
(2.3)z∈NCP(x)⟺∃σ>0,〈z,y-x〉≤σ‖y-x‖2,∀y∈C.Clarke et al. [7] and Poliquin et al. [8] have introduced and studied a new class of nonconvex sets, which are called uniformly prox-regular sets. This class of uniformly prox-regular sets has played an important part in many nonconvex applications such as optimization, dynamic systems, and differential inclusions.Definition 2.4.
For a givenr∈(0,+∞], a subset C of ℋ is said to be uniformly prox-regular with respect to r if, for all x¯∈C and for all 0≠z∈NCP(x¯), one has
(2.4)〈z∥z∥,x-x¯〉≤12r‖x-x¯‖2,∀x∈C.
We make the convention 1/r=0 for r=+∞.It is well known that a closed subset of a Hilbert space is convex if and only if it is proximally smooth of radiusr>0. Thus, in view of Definition 2.4, for the case of r=∞, the uniform r-prox-regularity C is equivalent to the convexity of C. Moreover, it is clear that the class of uniformly prox-regular sets is sufficiently large to include the class p-convex sets, C1,1 submanifolds (possibly with boundary) of ℋ, the images under a C1,1 diffeomorphism of convex sets, and many other nonconvex sets; see [6, 8].Now, let us state the following facts, which summarize some important consequences of the uniform prox-regularity. The proof of this result can be found in [7, 8].Lemma 2.5.
LetC be a nonempty closed subset of ℋ,r∈(0,+∞] and set Cr:={x∈ℋ;d(x,C)<r}. If C is uniformly r-uniformly prox-regular, then the following hold: (1)
for allx∈Cr, projC(x)≠∅,(2)
for alls∈(0,r), projC is Lipschitz continuous with constant r/(r-s) on Cs,(3)
the proximal normal cone is closed as a set-valued mapping.In this paper, we are interested in the following classes of nonlinear mappings.Definition 2.6.
A mappingT:C→ℋ is said to be (a)
γ-strongly monotone if there exists a constant γ>0 such that
(2.5)〈Tx-Ty,x-y〉≥γ‖x-y‖2,∀x,y∈C,(b)
μ-Lipschitzif there exist a constants μ>0 such that
(2.6)∥Tx-Ty∥≤μ∥x-y∥,∀x,y∈C.


## 3. System of Nonconvex Variational Inequalities Involving Nonmonotone Mapping

Letℋ be a real Hilbert space, and let C be a nonempty closed subset of ℋ. In this section, we will consider the following problem: find x*,y*∈C such that(3.1)y*-x*-ρTy*∈NCP(x*),x*-y*-ηTx*∈NCP(y*),
where ρ and η are fixed positive real numbers, C is a closed subset of ℋ, and T:C→ℋ is a mapping.The iterative algorithm for finding a solution of the problem (3.1) was considered by Moudafi [9], when C is r-uniformly prox-regular and T is a strongly monotone mapping. He also remarked that two-step models (3.1) for nonlinear variational inequalities are relatively more challenging than the usual variational inequalities since it can be applied to problems arising, especially from complementarity problems, convex quadratic programming, and other variational problems. In this section, we will generalize such result by considering the conditions for existence solution of problem (3.1) when T is not necessary strongly monotone. To do so, we will use the following algorithm as an important tool.Algorithm 3.1.
LetC be an r-uniformly prox-regular subset of ℋ. Assume that T:C→ℋ is a nonlinear mapping. Letting x0 be an arbitrary point in C, we consider the following two-step projection method:
(3.2)yn=projC[xn-η(Txn)],xn+1=projC[yn-ρ(Tyn)],
where ρ, η are positive reals number, which were appeared in problem (3.1).Remark 3.2.
The projection algorithm above has been introduced in the convex case, and its convergence was proved see [10]. Observe that (3.2) is well defined provided the projection on C is not empty. Our adaptation of the projection algorithm will be based on Lemma 2.5.Now we will prove the existence theorems of problem (3.1), when C is a closed uniformly r-prox-regular. Moreover, from now on, the number r will be understood as a finite positive real number (if not specified otherwise). This is because, as we know, if r=∞, then such a set C is nothing but the closed convex set.We start with an important remark.Remark 3.3.
LetC be a uniformly r-prox-regular closed subset of ℋ. Let T1,T2:C→ℋ be such that T1 is a μ1-Lipschitz continuous, γ-strongly monotone mapping and T2 is a μ2-Lipschitz continuous mapping. If ξ=r[(μ12-γμ2)2-μ12(γ-μ2)2μ12-γμ2-(μ12-γμ2)2-μ12(γ-μ2)2(μ12-γμ2)2-μ12(γ-μ2)2]/μ12, then for each s∈(0,ξ) we have
(3.3)γts-μ2>(μ12-μ22)(ts2-1),
where ts=r/(r-s).It is worth to point out that, in Remark3.3, we have to assume that μ2<μ1. Thus, from now on, without loss of generality we will always assume that μ2<μ1.Theorem 3.4.
LetC be a uniformly r-prox-regular closed subset of a Hilbert space ℋ, and let T:C→ℋ be a nonlinear mapping. Let T1,T2:C→ℋ be such that T1 is a μ1-Lipschitz continuous and γ-strongly monotone mapping, T2 is a μ2-Lipschitz continuous mapping. If T=T1+T2 and the following conditions are satisfied: (a)
Mρ,ηδT(C)<ξ, where δT(C)=sup{∥u-v∥;u,v∈T(C)};(b)
there existss∈(Mρ,ηδT(C),ξ) such that
(3.4)γts-μ2ts(μ12-μ22)-ζ<ρ,η<min{γts-μ2ts(μ12-μ22)+ζ,1tsμ2},where Mρ,η=max{ρ,η}, ts=r/(r-s), and ζ=(tsγ-μ2)2-(μ12-μ22)(ts2-1)/ts(μ12-μ22).
Then the problem (3.1) has a solution. Moreover, the sequence (xn,yn) which is generated by (3.2) strongly converges to a solution (x*,y*)∈C×C of the problem (3.1).Proof.
Firstly, by condition (b), we can easily check thatyn-ρTyn and xn-ηTxn belong to the set Cs, for all n=1,2,3,…. Thus, from Lemma 2.5 (1), we know that (3.2) is well defined. Consequently, from (3.2) and Lemma 2.5 (2), we have
(3.5)∥xn+1-xn∥=∥projC(yn-ρTyn)-projC(yn-1-ρTyn-1)∥≤ts∥yn-yn-1-ρ(Tyn-Tyn-1)∥≤ts[∥yn-yn-1-ρ(T1yn-T1yn-1)∥+ρ∥T2yn-T2yn-1∥].
Since the mapping T1 is γ-strongly monotone and μ1-Lipschitz continuous, we obtain
(3.6)‖yn-yn-1-ρ(T1yn-T1yn-1)‖2=‖yn-yn-1‖2-2ρ〈yn-yn-1,T1yn-T1yn-1〉+ρ2‖T1yn-T1yn-1‖2≤‖yn-yn-1‖2-2ργ∥yn-yn-1∥+ρ2μ12‖yn-yn-1‖2=(1-2ργ+ρ2μ12)‖yn-yn-1‖2.
On the other hand, since T2 is μ2-Lipschitz continuous, we have
(3.7)∥T2yn-T2yn-1∥≤μ2∥yn-yn-1∥.
Thus, by (3.5), (3.6), and (3.7), we obtain
(3.8)∥xn+1-xn∥≤ts[ρμ2+1-2ργ+ρ2μ12]∥yn-yn-1∥.
Similarly, we have
(3.9)∥yn-yn-1∥=∥projC(xn-ηTxn)-projC(xn-1-ηTxn-1)∥≤ts∥xn-xn-1-η(Txn-Txn-1)∥≤ts[∥xn-xn-1-η(T1xn-T1xn-1)∥+η∥T2xn-T2xn-1∥]≤ts[ημ2+1-2ηγ+η2μ12]∥xn-xn-1∥.
Combining (3.8) and (3.9), we get
(3.10)∥xn+1-xn∥≤ts2θρθη∥xn-xn-1∥,
where θρ:=ρμ2+1-2ργ+ρ2μ12 and θη:=ημ2+1-2ηγ+η2μ12. Moreover, by (3.4), we know that tsθρ and tsθη are elements of the interval (0,1). Thus, from (3.10), it follows that
(3.11)∥xn+1-xn∥≤κn∥x1-x0∥
for all n=1,2,3,…, where κ:=ts2θρθη. Hence, for any m≥n>1, it follows that
(3.12)∥xm-xn∥≤∑i=nm-1∥xi+1-xi∥≤∑i=nm-1κi∥x1-x0∥≤κn1-κ∥x1-x0∥.
Since κ<1, it follows that κn→0 as n→∞, and this implies that {xn}⊂C is a Cauchy sequence. Consequently, from (3.9), we also have that {yn} is a Cauchy sequence in C. Thus, by Lemma 2.5 (3), the closedness property of C implies that there exists (x*,y*)∈C×C such that (xn,yn)→(x*,y*) as n→∞.
We claim that(x*,y*)∈C×C is a solution of the problem (3.1). Indeed, by the definition of the proximal normal cone, from (3.2), we have
(3.13)(xn-yn)-η(Txn)∈NCP(yn),(yn-xn+1)-ρ(Tyn)∈NCP(xn+1).
By letting n→∞, using the closedness property of the proximal cone together with the continuity of T, we have
(3.14)x*-y*-η(Tx*)∈NCP(y*),y*-x*-ρ(Ty*)∈NCP(x*).
This completes the proof.Immediately, by settingT2=0, we have the following result.Theorem 3.5.
LetC be a uniformly r-prox-regular closed subset of a Hilbert space ℋ. Let T:C→ℋ be a μ-Lipschitz continuous and γ-strongly monotone mapping. If the following conditions are satisfied: (a)
Mρ,ηδT(C)<ξ, where δT(C)=sup{∥u-v∥;u,v∈T(C)};(b)
there existss∈(Mρ,ηδT(C),ξ) such that
(3.15)γμ2-ζ<ρ,η<γμ2+ζ,where ζ=(tsγ)2-(μ12)(ts2-1)/ts(μ12) and ts=r/(r-s).
Then the problem (3.1) has a solution. Moreover, the sequence (xn,yn) which is generated by (3.2) strongly converges to a solution (x*,y*)∈C×C of the problem (3.1).In view of proving Theorem3.4, we can obtain the following result, which contains a recent result presented by Moudafi [9] as a special case.Theorem 3.6.
LetC be a uniformly r-prox-regular closed subset of a Hilbert space ℋ, and let T:C→ℋ be a mapping. Let T1,T2:C→ℋ be such that T1 is a μ1-Lipschitz continuous and γ-strongly monotone mapping, T2 is a μ2-Lipschitz continuous mapping. If T=T1+T2 and there exists s∈(0,ξ) such that
(3.16)γts-μ2ts(μ12-μ22)-ζ<ρ<min{γts-μ2ts(μ12-μ22)+ζ,1tsμ2,s1+∥Tyn∥},γts-μ2ts(μ12-μ22)-ζ<η<min{γts-μ2ts(μ12-μ22)+ζ,1tsμ2,s1+∥Txn∥}
for all n=1,2,3,…, where ts=r/(r-s), ζ=(tsγ-μ2)2-(μ12-μ22)(ts2-1)/ts(μ12-μ22) and the sequence (xn,yn) was generated by (3.2), then the sequence (xn,yn) strongly converges to a solution (x*,y*)∈C×C of the problem (3.1).Remark 3.7.
(i) An inspection of Theorem3.6 shows that the sequences {Txn} and {Tyn} are bounded.
(ii) By settingT2=:0, we see that Theorem 3.6 reduces to a result presented by Moudafi [9].Remark 3.8.
IfC is a convex set, by the definition of the proximal normal cone, we can reformulate (3.1) as follows: find x*,y*∈C×C such that
(3.17)〈ρT(y*)+x*-y*,x-x*〉≥0,∀x∈C,〈ηT(x*)+y*-x*,x-y*〉≥0,∀x∈C.
The problem (3.17) was introduced and studied by Verma [10], when T is a strong monotone mapping. Hence, Theorem 3.4 extends and improves the results presented by Verma [10]. For further recent results related to the problem (3.17), see also [2, 3, 5, 11–13].


## 4. Further Results

By using the techniques as in Theorem3.4, we can also obtain an existence theorem of the following problem: find x*∈C such that(4.1)-Tx*∈NCP(x*).
The problem of type (4.1) was studied by Noor [14] but in a finite dimension Hilbert space setting. In this section, we intend to consider the problem (4.1) in an infinite dimension Hilbert space. To do this, the following remark is useful.Remark 4.1.
LetT:C→C be a γ-strongly monotone and μ-Lipschitz continuous mapping. Then, the function f:(1,μ2/(μ2-γ2))→(0,∞) which is defined by
(4.2)f(t)=t2(γ2-μ2)+μ2tμ2,∀t∈(1,μ2μ2-γ2),
is a continuous decreasing function on its domain.We now close this section by proving an existence theorem to the problem (4.1) in a nonconvex infinite dimensional setting.Theorem 4.2.
LetC be a uniformly r-prox-regular closed subset of a Hilbert space ℋ, and let T:C→ℋ be a γ-strongly monotone and μ-Lipschitz continuous mapping. If 0<δT(C)≤γr, then the problem (4.1) has a solution.Proof.
Firstly, by using an elementary calculation, we know that the functionh:[1,μ2/(μ2-γ2))→(0,∞) which is defined by
(4.3)h(t)=r(t-1)tδT(C)+f(t),∀t∈[1,μ2μ2-γ2),
is a continuous increasing function on [1,(μ2r2-δT(C)2)/r2(μ2-γ2)]. Moreover, we see that the net {ts}s∈(0,r) which is defined by ts=:r/(r-s) converges to 1 as s↓0. Using these observations, together with the fact that h(t)↓γ/μ2 as t↓1, we can find s*∈(0,r(r2γ2-δT(C)2)/(μ2r2-δT(C)2)) such that μ2h(ts*)>γ. It is worth to notice that, from the choice of s*, we have γ/μ2-f(ts*)<s*/δT(C).
Now, we choose a fixed positive real numberρ such that
(4.4)γμ2-f(ts*)<ρ<min{γμ2+f(ts*),s*δT(C)}.
Next, let us start with an element x0∈C and use an induction process to obtain a sequence {xn}⊂C satisfying
(4.5)xn+1=projC(xn-ρTxn),∀n=0,1,2,….
Note that, because of the choice of ρ, we can easily check that xn-ρTxn∈Cs* for all n=1,2,3,…. Following the proof of Theorem 3.4, we know that {xn} is a Cauchy sequence in C. If xn→x* as n→∞, the closedness property of the proximal cone together with the continuity of T, from (4.5), we see that x* is a solution of the problem (4.1). This completes the proof.Remark 4.3.
Theorems3.4, 3.5, and 4.2 not only give the conditions for the existence solution of the problems (3.1) and (4.1), respectively, but also provide the algorithm to find such solutions for any initial vector x0∈C.



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