# Approximation for the Hierarchical Constrained Variational Inequalities over the Fixed Points of Nonexpansive Semigroups

Authors: Li-Jun Zhu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a pointx* such that x*∈Ω,〈(A-γf)x*-(I-B)Sx*,x-x*〉≥0,∀x∈Ω, where Ω is the set of the solutions of the following variational inequality: x*∈Ϝ,〈(A-S)x*,x-x*〉≥0,∀x∈Ϝ, where A,B are two strongly positive bounded linear operators, f is a ρ-contraction, S is a nonexpansive mapping, and Ϝ is the fixed points set of a nonexpansive semigroup {T(s)}s≥0. We present a double-net convergence hierarchical to some elements in Ϝ which solves the above hierarchical constrained variational inequalities.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let C be a nonempty closed convex subset of H. Recall that a self-mapping f of C is said to be contractive if there exists a constant ρ∈[0,1) such that ∥f(x)-f(y)∥≤ρ∥x-y∥ for all x,y∈C. A mapping T:C→C is called nonexpansive if
(1)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.We denote byFix(T) the set of fixed points of T; that is, Fix(T)={x∈C:Tx=x}. A bounded linear operator B is called strongly positive on H if there exists a constant γ~>0 such that
(2)〈Bx,x〉≥γ~∥x∥2,∀x∈H.It is well known that the variational inequality for an operator,φ:H→H, over a nonempty, closed, and convex set, C⊂H, is to find a point x*∈C with the property
(3)〈φ(x*),y-x*〉≥0,∀y∈C.The set of the solutions of the variational inequality (3) is denoted by VI(C,φ). If the mapping φ is a monotone operator, then we say that VI(3) is monotone. It is well known that if φ is Lipschitzian and strongly monotone, then for small enough δ>0, the mapping PC(I-δφ) is a contraction on C and so the sequence {xn} of Picard iterates, given by xn=PC(I-δφ)xn-1(n≥1), converges strongly to the unique solution of the VI(3). This sort of VI(3) where φ is strongly monotone and Lipschitzian is originated from Yamada [1]. However, if φ is only monotone (not strongly monotone), then their iterative methods do not apply to VI.Many practical problems such as signal processing and network resource allocation are formulated as the variational inequality over the set of the solutions of some nonlinear mappings (e.g., the fixed point set of nonexpansive mappings), and algorithms to solve these problems have been proposed. Iterative algorithms have been presented for the convex optimization problem with a fixed point constraint along with proof that these algorithms strongly converge to the unique solution of problems with a strongly monotone operator. The strong monotonicity condition guarantees the uniqueness of the solution. For some related works on the variational inequalities, please see [2–23] and the references therein. Particulary, the variational inequality problems over the fixed points of nonexpansive mappings have been considered. The reader can consult [16, 24]. On the other hand, we note that in the literature, nonlinear ergodic theorems for nonexpansive semigroups have been considered by many authors; see, for example, [25–32]. In this paper, we will consider a general variational inequality problem with the variational inequality constraint is the fixed points of nonexpansive semigroups.The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a pointx* such that
(4)x*∈Ω,〈(A-γf)x*-(I-B)Sx*,x-x*〉≥0,∀x∈Ω,
where Ω is the set of the solutions of the following variational inequality:
(5)x*∈Ϝ,〈(A-S)x*,x-x*〉≥0,∀x∈Ϝ,
where A and B are two strongly positive bounded linear operators, f is a ρ-contraction, S is a nonexpansive mapping, and Ϝ is the fixed points set of a nonexpansive semigroup {T(s)}s≥0. We present a double-net convergence hierarchical to some elements in Ϝ which solves the above hierarchical constrained variational inequalities.


## 2. Preliminaries

LetC be a nonempty closed convex subset of a real Hilbert space H. The metric (or the nearest point) projection from H onto C is the mapping PC:H→C which assigns to each point x∈C the unique point PCx∈C satisfying the property
(6)∥x-PCx∥=infy∈C∥x-y∥=:d(x,C).It is well known thatPC is a nonexpansive mapping and satisfies
(7)〈x-y,PCx-PCy〉≥∥PCx-PCy∥2,∀x,y∈H.Moreover,PC is characterized by the following property:
(8)〈x-PCx,y-PCx〉≤0,∀x∈H,y∈C.Recall that a family{T(s)}s≥0 of mappings of H into itself is called a nonexpansive semigroup if it satisfies the following conditions:(S1)
T(0)x=x for all x∈H;(S2)
T(s+t)=T(s)T(t) for all s,t≥0;(S3)
∥T(s)x-T(s)y∥≤∥x-y∥ for all x,y∈H and s≥0;(S4)
for allx∈H, s→T(s)x is continuous.We denote byFix(T(s)) the set of fixed points of T(s) and by Ϝ the set of all common fixed points of {T(s)}s≥0; that is, Ϝ=⋂s≥0‍Fix(T(s)). It is known that Ϝ is closed and convex.We need the following lemmas for proving our main results.Lemma 1 (see [33]).
LetC be a nonempty bounded closed convex subset of a Hilbert space H and let {T(s)}s≥0 be a nonexpansive semigroup on C. Then, for every h≥0,
(9)limt→∞supx∈C∥1t∫0t‍T(s)xds-T(h)1t∫0t‍T(s)xds∥=0.Lemma 2 (see [34]).
LetC be a closed convex subset of a real Hilbert space H and let S:C→C be a nonexpansive mapping. Then, the mapping I-S is demiclosed. That is, if {xn} is a sequence in C such that xn→x* weakly and (I-S)xn→y strongly, then (I-S)x*=y.Lemma 3 (see [35]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Assume that a mapping F:C→H is monotone and weakly continuous along segments (i.e., F(x+ty)→F(x) weakly, as t→0, whenever x+ty∈C for x,y∈C). Then the variational inequality
(10)x*∈C,〈Fx*,x-x*〉≥0,∀x∈C,
is equivalent to the dual variational inequality
(11)x*∈C,〈Fx,x-x*〉≥0,∀x∈C.


## 3. Main Results

Now we consider the following hierarchical variational inequality with the variational inequality constraint over the fixed points set of nonexpansive semigroups{T(s)}s≥0.Problem 1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction with coefficient ρ∈[0,1) and letS:C→C be a nonexpansive mapping. Let {T(s)}s≥0 be a nonexpansive semigroup on C and let A,B:H→H be two strongly positive bounded linear operators with coefficients λ~(1≤λ~<2) and γ~(0<γ~<1), respectively. Let γ be a constant satisfying 0<γρ<γ~.
Now, our objective is to findx* such that
(12)x*∈Ω,〈(A-γf)x*-(I-B)Sx*,x-x*〉≥0,∀x∈Ω,
where Ω:=VI(Ϝ,A-S) is the set of the solutions of the following variational inequality:
(13)x*∈Ϝ,〈(A-S)x*,x-x*〉≥0,∀x∈Ϝ.We observe that(A-γf)-(I-B)S is strongly monotone and Lipschitz continuous. In fact, we have
(14)〈(A-γf)x-(I-B)Sx-[(A-γf)y-(I-B)Sy],x-y〉=〈Ax-Ay,x-y〉-γ〈f(x)-f(y),x-y〉-(I-B)〈Sx-Sy,x-y〉≥λ~∥x-y∥2-γρ∥x-y∥2-(1-γ~)∥x-y∥2=(λ~-1+γ~-γρ)∥x-y∥2,∥(A-γf)x-(I-B)Sx-[(A-γf)y-(I-B)Sy]∥≤∥Ax-Ay∥+γ∥f(x)-f(y)∥+∥I-B∥∥Sx-Sy∥≤∥A∥∥x-y∥+γρ∥x-y∥+(1-γ~)∥x-y∥=(1+∥A∥+γρ-γ~)∥x-y∥.Hence, the existence and the uniqueness of the solution to Problem1 are guaranteed.In order to solve the above hierarchical constrainted variational inequality, we present the following double net.Algorithm 4.
Setκ=1/(λ~+γ~-γρ). Then, 0<κ≤1. For each (s,t)∈(0,κ)×(0,1), we define a double net {xs,t} implicitly by
(15)xs,t=PC[∫0λs‍s(tγf(xs,t)+(I-tB)Sxs,t)+(I-sA)1λs∫0λs‍T(ν)xs,tdν].Note that this implicit manner algorithm is well defined. In fact, we define the mapping(16)x↦Ws,t(x):=PC[∫0λs‍s(tγf(x)+(I-tB)Sx)+(I-sA)1λs∫0λs‍T(ν)xdν],(s,t)∈(0,κ)×(0,1).Note that this self-mapping is a contraction. As a matter of fact, we have(17)∥Ws,t(x)-Ws,t(y)∥=∥PC[∫0λs‍s(tγf(x)+(I-tB)Sx)+(I-sA)1λs∫0λs‍T(ν)xdν]-PC[∫0λs‍s(tγf(y)+(I-tB)Sy)+(I-sA)1λs∫0λs‍T(ν)ydν]∥≤stγ∥f(x)-f(y)∥+s∥I-tB∥∥Sx-Sy∥+∥I-sA∥∥1λs∫0λs‍T(ν)xdν-1λs∫0λs‍T(ν)ydν∥≤[1-(λ~-1)s-(γ~-γρ)st]∥x-y∥.
Since (s,t)∈(0,κ)×(0,1), 0<1-(λ~-1)s-(γ~-γρ)st<1. Hence, Ws,t is a contraction. Therefore, by Banach’s Contraction Principle, Ws,t has a unique fixed point which is denoted by xs,t∈C.Next we show the behavior of the net{xs,t} as s→0 and t→0 successively.Theorem 5.
Assume thatVI(Ϝ,A-S)≠∅. Then, for each fixed t∈(0,1), the net {xs,t} defined by (15) converges in norm, as s→0+, to a solution xt∈Ϝ. Moreover, as t→0+, the net {xt} converges in norm to the unique solution x* of Problem 1.Proof.
We first show that the sequence{xs,t} is bounded. Take y*∈Ϝ. From (15), we have
(18)∥xs,t-y*∥=∥∫0λs‍Pc[∫0λs‍s(tγf(xs,t)+(I-tB)Sxs,t)+(I-sA)1λs∫0λs‍T(ν)xdν]-y*∥≤∥∫0λs‍s(tγf(xs,t)+(I-tB)Sxs,t)+(I-sA)1λs∫0λs‍T(ν)xdν-y*∥=∥∫0λs‍stγ(f(xs,t)-f(y*))+s(I-tB)(Sxs,t-Sy*)+stγf(y*)+s(I-tB)Sy*-sAy*+(I-sA)(1λs∫0λs‍T(ν)xdν-y*)∥≤stγ∥f(xs,t)-f(y*)∥+s∥I-tB∥∥Sxs,t-Sy*∥+∥I-sA∥×∥1λs∫0λs‍T(ν)xdν-y*∥+s∥tγf(y*)+(I-tB)Sy*-Ay*∥≤stγρ∥xs,t-y*∥+s(1-tγ~)∥xs,t-y*∥+(1-sλ~)∥xs,t-y*∥+s∥tγf(y*)+(I-tB)Sy*-Ay*∥=[1-(λ~-1)s-(γ~-γρ)st]∥xs,t-y*∥+s∥tγf(y*)+(I-tB)Sy*-Ay*∥.
Hence(19)∥xs,t-y*∥≤1(γ~-γρ)t+λ~-1×∥tγf(y*)+(I-tB)Sy*-Ay*∥.
It follows that for each fixed t∈(0,1), {xs,t} is bounded.
Next, we show thatlims→0∥T(τ)xs,t-xs,t∥=0 for all 0≤τ<∞ and consequently, as s→0+, the entire net {xs,t} converges in norm to xt∈Ϝ.
For each fixedt∈(0,1), we set Rt:=(1/((γ~-γρ)t+λ~-1))∥tγf(y*)+(I-tB)Sy*-Ay*∥. It is clear that for each fixed t∈(0,1), {xs,t}⊂B(y*,Rt). Notice that
(20)∥1λs∫0λs‍T(ν)xs,tdν-y*∥≤∥xs,t-y*∥≤Rt.
Moreover, we observe that ifx∈B(y*,Rt), then
(21)∥T(s)x-y*∥≤∥T(s)x-T(s)y*∥≤∥x-y*∥≤Rt,
that is,B(y*,Rt) is T(s)-invariant for all s.
From (15), we deduce
(22)∥T(τ)xs,t-xs,t∥≤∥T(τ)xs,t-T(τ)1λs∫0λs‍T(ν)xs,tdν∥+∥T(τ)1λs∫0λs‍T(ν)xs,tdν-1λs∫0λs‍T(ν)xs,tdν∥+∥1λs∫0λs‍T(ν)xs,tdν-xs,t∥≤∥T(τ)1λs∫0λs‍T(ν)xs,tdν-1λs∫0λs‍T(ν)xs,tdν∥+2∥xs,t-1λs∫0λs‍T(ν)xs,tdν∥≤2s∥∫0λs‍tγf(xs,t)+(I-tB)Sxs,t-Aλs∫0λs‍T(ν)xs,tdν∥+∥T(τ)1λs∫0λs‍T(ν)xs,tdν-1λs∫0λs‍T(ν)xs,tdν∥.
Since{xs,t} is bounded, {f(xs,t)} and {Sxs,t} are also bounded. Then, from Lemma 1, we deduce for all 0≤τ<∞ and fixed t∈(0,1) that
(23)lims→0∥T(τ)xs,t-xs,t∥=0.
Setys,t=s(tγf(xs,t)+(I-tB)Sxs,t)+(I-sA)(1/λs)∫0λs‍T(ν)xs,tdν for all (s,t)∈(0,κ)×(0,1). We then have xs,t=PCys,t, and for any y*∈Ϝ,
(24)xs,t-y*=xs,t-ys,t+ys,t-y*=xs,t-ys,t+s(tγf(xs,t)+(I-tB)Sxs,t)+(I-sA)1λs∫0λs‍T(ν)xs,tdν-y*=xs,t-ys,t+stγ(f(xs,t)-f(y*))+s(I-tB)(Sxs,t-Sy*)+s(tγf(y*)+(I-tB)Sy*-Ay*)+(I-sA)(1λs∫0λs‍T(ν)xs,tdν-y*).
Notice that(25)〈xs,t-ys,t,xs,t-y*〉≤0.
Thus, we have(26)∥xs,t-y*∥2=〈xs,t-ys,t,xs,t-y*〉+stγ〈f(xs,t)-f(y*),xs,t-y*〉+s(I-tB)〈Sxs,t-Sy*,xs,t-y*〉+(I-sA)×〈1λs∫0λs‍T(ν)xs,tdν-y*,xs,t-y*〉+s〈tγf(y*)+(I-tB)Sy*-Ay*,xs,t-y*〉≤stγ∥f(xs,t)-f(y*)∥∥xs,t-y*∥+s∥I-tB∥∥Sxs,t-Sy*∥∥xs,t-y*∥+s〈tγf(y*)+(I-tB)Sy*-Ay*,xs,t-y*〉+∥I-sA∥∥1λs∫0λs‍T(ν)xs,tdν-y*∥∥xs,t-y*∥≤stγρ∥xs,t-y*∥2+s(1-tγ~)∥xs,t-y*∥2+s〈tγf(y*)+(I-tB)Sy*-Ay*,xs,t-y*〉+(1-sλ~)∥xs,t-y*∥2=[1-(λ~-1)s-(γ~-γρ)st]∥xs,t-y*∥2+s〈tγf(y*)+(I-tB)Sy*-Ay*,xs,t-y*〉.
So(27)∥xs,t-y*∥2≤1(γ~-γρ)t+λ~-1×〈tγf(y*)+(I-tB)Sy*-Ay*,xs,t-y*〉,y*∈Ϝ.
Assume{sn}⊂(0,1) such that sn→0 as n→∞. By (27), we obtain immediately that
(28)∥xsn,t-y*∥2≤1(γ~-γρ)t+λ~-1×〈tγf(y*)+(I-tB)Sy*-Ay*,xsn,t-y*〉,y*∈Ϝ.
Since{xsn,t} is bounded, there exists a subsequence {sni} of {sn} such that {xsni,t} converges weakly to a point xt. From (23) and Lemma 2, we get xt∈Ϝ. We can substitute xt for y* in (28) to get
(29)∥xsn,t-xt∥2≤1(γ~-γρ)t+λ~-1×〈tγf(xt)+(I-tB)Sxt-Axt,xsn,t-xt〉.
The weak convergence of{xsn,t} to xt actually implies that xsn,t→xt strongly. This has proved the relative norm-compactness of the net {xs,t} as s→0+ for each fixed t∈(0,1).
In (28), we take the limit as n→∞ to get
(30)∥xt-y*∥2≤1(γ~-γρ)t+λ~-1×〈tγf(y*)+(I-tB)Sy*-Ay*,xt-y*〉,∀y*∈Ϝ.
In particular,xt solves the following variational inequality:
(31)xt∈Ϝ,〈Ay*-tγf(y*)-(I-tB)Sy*,y*-xt〉≥0,∀y*∈Ϝ.
Note that the mappingA-tγf-(I-tB)S is monotone for all t∈(0,1), since
(32)〈Ax-tγf(x)-(I-tB)Sx-(Ay-tγf(y)-(I-tB)Sy),x-y〉=〈Ax-Ay,x-y〉-tγ〈f(x)-f(y),x-y〉-(I-tB)〈Sx-Sy,x-y〉≥λ~∥x-y∥2-tγρ∥x-y∥2-(1-tγ~)∥x-y∥2=[λ~-1+(γ~-γρ)t]∥x-y∥2≥0.
By Lemma3, (31) is equivalent to its dual VI:
(33)xt∈Ϝ,〈Axt-tγf(xt)-(I-tB)Sxt,y*-xt〉≥0,∀y*∈Ϝ.
Next we show that ass→0+, the entire net {xs,t} converges in norm to xt∈Ϝ. We assume xsn′,t→xt′, where sn′→0. Similarly, by the above proof, we deduce xt′∈Ϝ which solves the following variational inequality:
(34)xt′∈Ϝ,〈Axt′-tγf(xt′)-(I-tB)Sxt′,y*-xt′〉≥0,∀y*∈Ϝ.
In (33), we take y*=xt′ to get
(35)〈Axt-tγf(xt)-(I-tB)Sxt,xt′-xt〉≥0.
In (34), we take y*=xt to get
(36)〈Axt′-tγf(xt′)-(I-tB)Sxt′,xt-xt′〉≥0.
Adding up (35) and (36) yields
(37)〈Axt-Axt′,xt-xt′〉-tγ〈f(xt)-f(xt′),xt-xt′〉-(I-tB)〈Sxt-Sxt′,xt-xt′〉≤0.
At the same time we note that(38)〈Axt-Axt′,xt-xt′〉-tγ〈f(xt)-f(xt′),xt-xt′〉-(I-tB)〈Sxt-Sxt′,xt-xt′〉≥[λ~-1+(γ~-γρ)t]∥xt-xt′∥2≥0.
Therefore, by (37) and (38), we deduce
(39)xt′=xt.Hence the entire net {xs,t} converges in norm to xt∈Ϝ as s→0+.
Ast→0+, the net {xt} converges to the unique solution x* of Problem 1.
In (33), we take any y*∈Ω to deduce
(40)〈Axt-tγf(xt)-(I-tB)Sxt,y*-xt〉≥0.
By virtue of the monotonicity ofA-S and the fact that y*∈Ω, we have
(41)〈Axt-Sxt,y*-xt〉≤〈Ay*-Sy*,y*-xt〉≤0.
We can rewrite (40) as
(42)〈t[Axt-γf(xt)-(I-B)Sxt]+(1-t)(Axt-Sxt),y*-xt〉≥0.
It follows from (41) and (42) that
(43)〈Axt-γf(xt)-(I-B)Sxt,y*-xt〉≥0,∀y*∈Ω.
Hence(44)∥xt-y*∥2≤〈xt-y*,xt-y*〉+〈γf(xt)+(I-B)Sxt-Axt,xt-y*〉=γ〈f(xt)-f(y*),xt-y*〉+(I-B)〈Sxt-Sy*,xt-y*〉+〈Ay*-Axt,xt-y*〉+〈γf(y*)+(I-B)Sy*-Ay*,xt-y*〉≤γρ∥xt-y*∥2+(1-γ~)∥xt-y*∥2-λ~∥xt-y*∥2+〈γf(y*)+(I-B)Sy*-Ay*,xt-y*〉=[1-(λ~+γ~-γρ)]∥xt-y*∥2+〈γf(y*)+(I-B)Sy*-Ay*,xt-y*〉.
Therefore,(45)∥xt-y*∥2≤1λ~+γ~-γρ×〈γf(y*)+(I-B)Sy*-Ay*,xt-y*〉,y*∈Ω.
In particular,(46)∥xt-y*∥≤1λ~+γ~-γρ×∥γf(y*)+(I-B)Sy*-Ay*∥,∀t∈(0,1),
which implies that{xt} is bounded.
We next prove thatωw(xt)⊂Ω; namely, if (tn) is a null sequence in (0,1) such that xtn→x′ weakly as n→∞, then x′∈Ω. To see this, we use (33) to get
(47)〈(A-S)xt,y*-xt〉≥t1-t〈γf(xt)+(I-B)Sxt-Axt,y*-xt〉,y*∈Ϝ.
However, sinceA-S is monotone,
(48)〈(A-S)y*,y*-xt〉≥〈(A-S)xt,y*-xt〉.
Combining the last two relations yields(49)〈(A-S)y*,y*-xt〉≥t1-t〈γf(xt)+(I-B)Sxt-Axt,y*-xt〉,y*∈Ϝ.
Lettingt=tn→0+ as n→∞ in (49), we get
(50)〈(A-S)y*,y*-x′〉≥0,y*∈Ϝ.
The equivalent dual VI of (50) is
(51)〈(A-S)x′,y*-x′〉≥0,y*∈Ϝ.
Namely,x′ is a solution of VI(13); hence x′∈Ω.
We further prove thatx′=x*, the unique solution of VI(12). As a matter of fact, we have by (45)
(52)∥xtn-x′∥2≤1λ~+γ~-γρ×〈γf(x′)+(I-B)Sx′-Ax′,xtn-x′〉,x′∈Ω.
Therefore, the weak convergence tox′ of {xtn} right implies that xtn→x′ in norm. Now we can let t=tn→0 in (45) to get
(53)〈γf(y*)+(I-B)Sy*-Ay*,y*-x′〉≤0,∀y*∈Ω,
which is equivalent to its dual VI(54)〈γf(x′)+(I-B)Sx′-Ax′,y*-x′〉≤0,∀y*∈Ω.
It turns out thatx′∈Ω solves VI(12). By uniqueness, we have x′=x*. This is sufficient to guarantee that xt→x* in norm, as t→0+. This completes the proof.Corollary 6.
For each(s,t)∈(0,κ)×(0,1), let {xs,t} be a double net defined by
(55)xs,t=PC[∫0λs‍s(tγf(xs,t)+(1-tB)Sxs,t)+(1-s)1λs∫0λs‍T(ν)xs,tdν],
for all(s,t)∈(0,κ)×(0,1). Then, for each fixed t∈(0,1), the net {xs,t} defined by (55) converges in norm, as s→0+, to a solution xt∈Ϝ. Moreover, as t→0+, the net {xt} converges in norm to x* which solves the following variational inequality:
(56)x*∈Ω,〈(I-γf)x*-(I-B)Sx*,x-x*〉≥0,∀x∈Ω,
where Ω is the set of the solutions of the following variational inequality:
(57)x*∈Ϝ,〈(I-S)x*,x-x*〉≥0,∀x∈Ϝ.Corollary 7.
For each(s,t)∈(0,κ)×(0,1), let {xs,t} be a double net defined by
(58)xs,t=PC[s((1-t)Sxs,t)+(1-s)1λs∫0λs‍T(ν)xs,tdν],
for all(s,t)∈(0,κ)×(0,1). Then, for each fixed t∈(0,1), the net {xs,t} defined by (58) converges in norm, as s→0+, to a solution xt∈Ϝ. Moreover, as t→0+, the net {xt} converges to the minimum norm solution x* of the following variational inequality:
(59)x*∈Ϝ,〈(I-S)x*,x-x*〉≥0,∀x∈Ϝ.Proof.
In (55), we take f=0 and B=I. Then (55) reduces to (58). Hence, the net {xt} defined by (58) converges in norm to x*∈Ω which satisfies
(60)x*∈Ω,〈x*,x-x*〉≥0,∀x∈Ω.
This indicates that(61)∥x*∥2≤〈x*,x〉≤∥x*∥∥x∥,∀x∈Ω.
Therefore,x* is the minimum norm solution of the VI(59). This completes the proof.



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*Source: 604369-2013-04-03.xml*