# Rate of Convergence of Tikhonov Method of Regularization for Constrained Linear Equations with Operators Having Closed Ranges

Authors: Milojica Jaćimović; Izedin Krnić; Oleg Obradović
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

We derive the estimates of the rate of convergence of the Tikhonov method of regularization for a constrained operator linear equation. In case that the range of the corresponding operator is closed, the estimate is of the same order as the estimates for a linear equation without constraints.


---

## Body

## 1. Introduction

LetH and F be real Hilbert spaces, A:H↦F—a linear bounded operator from H to F, f∈F—a fixed element, and U⊆H—a closed convex set. We will deal with the equation
(1)Au=f,u∈U.
Even when the constraint u∈U is absent, this problem can be ill-posed; that is, it is possible that there is u~ which is far from the set of solution of (1), such that Au~≈f. The ill-posedness of the equation in case of infinite-dimensional spaces H and F obviously comes from the fact that the range R(A):={Au:u∈H} of the operator A is nonclosed. However, if the operator A is known only approximately, then this equation can be ill-posed even in case of R(A)=R(A)¯. In this case, in order to solve the given problem, one has to use methods of regularization, for example, the Tikhonov method of regularization (see [1, 2]). Note that the presence of the constraints significantly complicates all issues concerning the ill-posedness.Let us suppose that the setU is given by
(2)U={u∈H:h(u)≤0},
where h:H↦R is a continuously differentiable convex function satisfying Lipschitz and Slater conditions:
(3)(∃L>0)(∀u,υ∈H)∥h′(u)-h′(υ)∥≤L∥u-υ∥,(4)∃u-∈U:h(u-)<0.We also suppose that setsU* and U∞ of the solutions of (1), and the corresponding equation without constraint
(5)Au=f,u∈H,
are nonempty. In this case, the minimization problems
(6)∥u∥2↦inf,u∈U*,(7)∥u∥2↦inf,u∈U∞
have unique solutions that will be denoted by u* and u∞ and called normal solutions of (1) and (5).Let us present one prominent example of the equation with the previously mentioned constraint with infinite-dimensional spacesH and/or F.Given thatT>0 and matrices C(·)=(cij(·))m×m, D(·)=(dij(·))m×q and G(·)=(gij(·))n×m, whose elements cij(·), dij(·), and gij(·) are piecewise continuous functions on [0,T]. Let us consider a system in the so-called state-variable (or state-space) form:
(8)x′(t)=C(t)x(t)+D(t)u(t),t∈[0,T],x(0)=0,y(t)=G(t)x(t),
where u=u(t),t∈[0,T] is a control from the space L2q[0,T]. This system of control is usually analyzed under some constraints on the control or on the trajectory. For example, the problems, “given f∈Rm, find a control
(9)u=u(·)∈U={u∈L2q:∫0T‍|x′(t,u(t))|2dt≤r2},
such that y(T)=y(T;u)=f” is a problem of type (1), (2). Then, one has to solve the equation Au=f on the set
(10)U={∫0T‍u∈H:h(u)=∫0T‍|x′(t,u(t))|2dt-r2≤0},
where
(11)Au=y(T;u):H=L2q[0,T]↦F=Rn.For example, electrical networks with a finite number of interconnected resistors, capacitors, and inductors can be described in state-variable form (see [3]).The problems of the type (1) in the literature are usually considered under the assumption that, instead of the exact operator A and instead of the element f, one actually deals with their approximations Aη∈ℒ(H,F),fδ∈F, such that
(12)∥A-Aη∥≤η,∥f-fδ∥≤δ,∥Aη∥≤a,
where η>0,δ>0 are small positive real numbers and a>0.As for the approximations of the normal solutionsu* and u∞, one can use [1, 4] the so-called Tikhonov regularized approximate solutions uα and υα, that is, the unique solutions of the regularized problems find uα∈U such that
(13)〈Aη*Aηuα-Aη*fδ+αuα,u-uα〉≥0,∀u∈U,(14)Aη*Aηυα+αυα=Aη*fδ.Let us observe that the above variational inequality (13) is necessary and sufficient conditions for the minimization problem of the Tikhonov function
(15)Tα(u)=12∥Aηu-fδ∥2+α2∥u∥2
on the set U or on the space H. The unique point uα(=uα(δ,η)) of the minimum of the function Tα on the set U is said to be a regularized solution of (1).The main aim of this paper is to establish whether there exists a choice of a parameter of regularizationα=α(δ,η) which implies the convergence rate O(δ+η) of regularized solutions uα=uα(δ,η) to u*. Namely, in [5], such convergence was proved in case of (5) without constraints and under the assumption that R(A)¯=R(A). Note also that even in case of H=R2, the rate of convergence uα(δ,η)→u*, depending on the boundary of the set U, can be arbitrarily slow (see [6]).Consequently, even in case ofR(A)=R(A)¯, only if we require additional conditions on h and u*, we can eventually expect the convergence rate O(δ+η) of regularized solutions uα to normal solution u*. In [7], it was proved that if u∞ and projection of h′(u*) belong to R(A*), then for α=O(δ+η) yields ∥uα-u*∥=O(δ+η)1/2.The estimates of the rate of convergence of regularized solutionsuα to the normal solution u* have been obtained in [8], under the conditions related to the smoothness of the boundary of the set U and of the properties of the projection of u* onto linear subspace of codimension 1, which is parallel to the tangential plane to the boundary of U.Let us emphasize that the results of this paper are inspired by the results from [8]. However, our results related to the rate of convergence of regularized solutions were obtained under somewhat different general conditions and under the assumptions that, instead of the exact operator A, and instead of the element f, we only know their approximations Aη and fδ.The paper is organized as follows.In Section2, we described Tikhonov method of regularization adapted to the case when the set of constraints is given by (2). This constraint is taken into account by Lagrange multipliers. Lemma 1 was proved under the conditions of the sourcewise representativity type, that is, under the assumption that u* belongs to the projection of R(A*)¯ onto U [2, 6, 8]. In this section, we also present some results concerning the properties of the operators Aη*Aη and AηAη*.In Section3, we suppose that the parameter of regularization α is chosen in such way that α=O(δ+η). Under this assumption, we derive the estimates of the rate of convergence of Lagrange multipliers of regularized problems. The main result of the paper is contained in Theorem 11 which refers to the rate of convergence of regularized solutions.


## 2.  Constrained Tikhonov Regularization and Previous Results

By applying the Kuhn-Tucker theorem to the minimization problem(16)Tα(u)=12∥Aηu-fδ∥2+α2∥u∥2,h(u)≤0,
we obtain that, for its unique solution uα=uα(δ,η), there is λα≥0 such that
(17)Aη*Aηuα-Aη*fδ+αuα+λαh′(uα)=0,hhhhhhhhhλαh(uα)=0.
Using the function gα(t)=1/(t+α),t≥0, (in what follows, we will use this notation) the first equality in (17) can be written as
(18)gα-1(Aη*Aη)uα-Aη*fδ+λαh′(uα)=0,
or as
(19)uα=vα-λαgα(Aη*Aη)h′(uα),
where vα=gα(Aη*Aη)Aη*fδ is the point of the minimum of the Tikhonov function Tα on H.In what follows, we assume that the parameterα is chosen such that α=O(δ+η).Note that ifh(vα)≤0, then u*=u∞, and for such α, the corresponding estimates concerning the accuracy of the regularized solutions of the equation Au=f,  u∈H, can be found, for example, in [5]. (Note that in [5], page 100, it was proved that if R(A)=R(A)¯, then for α=O(δ+η), the following estimate holds: ∥u∞-vα∥=O(δ+η)). For this reason, in what follows, we will suppose that h(vα)>0 for all sufficiently small δ and η. In this case, we have λα≠0 and, consequently, h(uα)=0 and h′(uα)≠0 for all such δ and η. Therefore, bearing in mind that uα→u* as δ,η→0, we conclude that h(u*)=0. The Slater condition (4) then implies that u* is not a point of minimum of h, and, consequently, h′(u*)≠0.Now, as a consequence of the variational inequality (13), taking into account the equality (17) and λα>0, we obtain
(20)(∀u∈U)〈h′(uα),u-uα〉≤0.
From here, we have
(21)(∀u∈U)〈h′(u*),u-u*〉≤0.
Consequently,
(22)〈h′(u*),uα-u*〉≤0,〈h′(uα),u*-uα〉≤0.Further estimates of the rate of convergence of the regularized solutions are based on the next property of the normal solution(23)u∞∈R(A*)¯=R(A*A)¯,
that for the operator with closed range becomes
(24)u∞∈R(A*A);thatis,u∞=R(A*A)w*,w*∈H.
As it was argued in [6] (see also [9]), this condition does not have the same meaning as it has in case of problem (6), because the normal solution u* of the corresponding problem without constraints does not satisfy the condition u*∈R(A*A). However, it is easy to prove that
(25)u*∈πU(R(A*))¯,
where by πX, we denote the operator of the metric projecting on the closed convex set X⊆H.Lemma 1.
u
*
∈
π
U
(
R
(
A
*
)
)
¯if and only if there exists γ*≥0 such that u*=u∞-γ*(I-πR(A*A)¯)h′(u*).Proof.
For anyu*∈πU(R(A*)¯), there is e*∈R(A*)¯ such that u* is a solution of the minimization problem
(26)12∥u-e*∥2⟶inf,h(u)≤0.
According to the Kuhn-Tucker theorem, there exists a real number γ*≥0 such that u*-e*+γ*h′(u*)=0. From here, bearing in mind that e*∈R(A*)¯, we obtain
(27)(I-πR(A*)¯)u*=-γ*(I-πR(A*)¯)h′(u*).
Consequently,
(28)u*=u∞+(I-πR(A*)¯)u*=u∞-γ*(I-πR(A*)¯)h′(u*).
Now, let us suppose that there existsγ*≥0 such that
(29)u*=u∞-γ*(I-πR(A*)¯)h′(u*).
This equality can be written in the form of(30)u*-(u∞-γ*πR(A*)¯h′(u*))=-γ*h′(u*).
Multiplying both sides of this equality by u-u* and bearing in mind (21), it comes out that
(31)〈u*-(u∞+γ*πR(A*)¯h′(u*)),u-u*〉=-γ*〈h′(u*),u-u*〉≥0,u∈U.
The last equality means that u*=πU(u∞+γ*πR(A*)¯h′(u*)). Having in mind that u∞+γ*πR(A*)¯h′(u*)∈R(A*)¯, it follows that u*∈πU(R(A*)¯).As an immediate consequence of this lemma, we have that if the range of the operatorA:H→F is closed subspace of F, then u*∈πU(R(A*)) if and only if there is γ*≥0such that u*=u∞-γ*(I-πR(A*))h′(u*).Finally, let us remark that it is easy to construct an example of convex setU⊆R2 and operator A:R2↦R2 such that u*∉πU(R(A*)).Further, we will continue with three lemmas related to properties of the operatorsAη*Aη and AηAη*, whose proofs can be found in [5], pages 99, 153, 154, and 156.Lemma 2.
If∥A-Aη∥≤η, then
(32)(gα(Aη*Aη)Aη*Aη-I)h⟶0asα⟶0,η⟶0
for all h∈R(A*)¯.Lemma 3.
(a) The range of the bounded linear operatorA:H→F is closed subspace of H if and only if
(33)mA:=inf{∥Au∥:∥u∥=1,u⊥KerA}>0.
(b) If the bounded linear operatorsA:H↦F and Aη:H↦F satisfy the condition
(34)mA>0,∥A-Aη∥≤η,0<2η<mA,
then σ(Aη*Aη),σ(AηAη*)⊆[0,η2]⋃‍[(mA-η)2,∥Aη∥2], where σ(Aη*Aη) and σ(AηAη*) are the spectrums of the operators Aη*Aη, and AηAη*.Byπη, we will denote the operator of orthogonal projecting of the space H on invariant subspace of the operator Aη*Aη that corresponds to the part of the spectrum belonging to [(mA-η)2,∥Aη∥2]. Then, I-πη is the operator of orthogonal projecting of the space H on invariant subspace of the operator Aη*Aη that corresponds to the part of the spectrum belonging to [0,η2].Lemma 4.
If the conditions (34) are satisfied, then,
(35)∥πR(A*)-πη∥≤ηmA-η,∥πη(I-πR(A*)¯)∥≤ηmA-η,∥Aη(I-πη)∥≤η.As the consequences of the previous lemmas, it is easy to prove the following estimates.Lemma 5.
Let the conditions (34) be satisfied. Then
(36)∥πηgα(Aη*Aη)∥≤1(mA-η)2.
If, additionally, the parameter α of regularization is chosen such that α=O(δ+η), then the following estimates hold for all x:
(37)∥gα(Aη*Aη)Aη*∥≤max{1;1mA-η},(38)∥(I-αgα(Aη*Aη))(I-πη)x∥≤max{1;1mA-η}∥(I-πη)x∥η,(39)∥αgα(Aη*Aη)x-(I-πη)x∥≤(1(mA-η)2+max{1;1mA-η})∥x∥(δ+η).The last result that we will cite in this section concerns the rate of convergence of the Tikhonov regularization of the equationAu=f, u∈H, and its proof can be found in [4] and also in [5].Theorem 6.
If the conditions (12) are satisfied and the parameter α=α(δ,η) is chosen such that α(δ,η)→0 and δ+η/α→0 as δ→0,η→0, then the regularized solutions vα(δ,η) of (14) converge to the normal solution u∞ of (5) as δ→0,η→0.
If, additionally, the range of the operatorA is closed, then for α=O(δ+η), the following estimate holds:
(40)∥vα-u∞∥=O(δ+η).


## 3. Convergence and Rate of Convergence

In order to derive an estimate for the solutions of the equationAu=f,u∈U⊆H, we will begin this section with lemmas concerning the properties of Lagrange multipliers λα for problem (17), that will be used in the proof of the estimates of the rate of convergence. Note that throughout the section we will suppose that the parameter of the regularization α is chosen such that α=O(δ+η).Lemma 7.
If the conditions (34), (12) are satisfied and (I-π)h′(u*)≠0, then for α=O(δ+η), the following estimate is true:
(41)|λαα-γ*|=O(δ+η+∥u*-uα∥),
where λα and γ* are Lagrange multipliers from (17) and Lemma 2.Proof.
By (39) and Lemma 4, from the equality
(42)αgα(Aη*Aη)h′(u*)-(I-πR(A*)¯)h′(u*)=αgα(Aη*Aη)h′(u*)-(I-πη)h′(u*)+(πR(A*)¯-πη)h′(u*),
we get that
(43)αgα(Aη*Aη)h′(u*)⟶(I-πR(A*)¯)h′(u*)hhhhhhhasδ,η⟶0,α=O(δ+η).
Furthermore, since uα→u*,h′(uα)→h′(u*), we have
(44)〈αgα(Aη*Aη)h′(u*),h′(uα)〉⟶∥(I-πR(A*)¯)h′(u*)∥2≠0hhhhhhhasα=O(δ+η)⟶0.
Consequently,
(45)〈αgα(Aη*Aη)h′(u*),h′(uα)〉≠0
for all sufficiently small δ and η. Multiplying both sides of (19) by h′(u*) and taking into consideration the last inequality, we obtain
(46)λαα=〈h′(u*),vα-uα〉〈αgα(Aη*Aη)h′(u*),h′(uα)〉.
Now, multiplying the equality u*=u∞-γ*(I-πR(A*)¯)h′(u*) (see Lemma 1) by (I-πR(A*)¯)h′(u*) and taking into account that u∞-u*∈KerA, we find that
(47)γ*=〈h′(u*),u∞-u*〉∥(I-πR(A*)¯)h′(u*)∥2.
In this way, we obtain the equality
(48)γ*-λαα=〈h′(u*),u∞-vα〉+〈h′(u*),uα-u*〉∥(I-πR(A*)¯)h′(u*)∥2+〈h′(u*),vα-uα〉∥(I-πR(A*)¯)h′(u*)∥2〈αgα(Aη*Aη)h′(u*),h′(uα)〉×[〈αgα(Aη*Aη)h′(u*),h′(uα)〉-∥(I-πR(A*)¯)u*∥2].
Let us observe that
(49)〈αgα(Aη*Aη)h′(u*),h′(uα)〉-∥(I-πR(A*)¯)u*∥2=〈αgα(Aη*Aη)h′(u*),h′(uα)-h′(u*)〉+〈αgα(Aη*Aη)g′(u*)-(I-πη)h′(u*),g′(u*)〉+〈(πR(A*)¯-πη)h′(u*),h′(u*)〉.
Now, having in mind condition (3), estimate (39), Lemma 4, and Theorem 6, we get
(50)|γ*-λαα|=O(δ+η+∥u*-uα∥).In the following theorem, we will present one result concerning the estimates of the discrepancies∥Auα-f∥ and ∥Aηuα-fδ∥.Theorem 8.
If the conditions (34) and (12) are satisfied and u*∈πU(R(A*)), then for α=O(δ+η), the following estimates hold:
(51)∥A(u*-uα)∥=O(δ+η),∥Aη(u*-uα)∥=O(δ+η),∥Aηuα-fδα∥=O(1).Proof.
First, let us consider the case when(I-πR(A*)¯)h′(u*)=0. Then, based on (19) and Lemma 1, we have that u*=u∞ and
(52)u*-uα=u∞-vα+λαgα(Aη*Aη)h′(uα).
Multiplying both sides of this equality by gα-1(Aη*Aη)(u*-uα) and taking into account (22), we obtain
(53)〈gα-1(Aη*Aη)(u*-uα),u*-uα〉≤〈gα-1(Aη*Aη)(u∞-vα),u*-uα〉.
The last inequality can be written in the form
(54)∥Aη(u*-uα)∥2+α∥u*-uα∥2≤∥Aη(u∞-vα)∥2+α∥u∞-vα∥2.
In this case, from here and from Theorem 6, it follows that
(55)∥Aη(u*-uα)∥=O(δ+η),∥A(u*-uα)∥≤∥(A-Aη)(u*-uα)∥+∥Aη(u*-uα)∥=O(δ+η).
Now, let us consider the case of(I-πR(A*)¯)h′(u*)≠0. Then, we have that u*=u∞-γ*(I-πR(A*)¯)h′(u*); that is
(56)u*-uα=u∞-vα+λαgα(Aη*Aη)h′(uα)-γ*(I-πR(A*)¯)h′(u*),
from where we obtain that
(57)Aη(u*-uα)=Aη(u∞-vα)+αλααAηgα(Aη*Aη)h′(uα)-γ*(Aη-A)×(I-πR(A*)¯)h′(u*).
From here, by Theorem 6, Lemma 5 and having in mind (37) and condition (12), we obtain the estimate
(58)∥Aη(u*-uα)∥=O(δ+η),∥A(u*-uα)∥=O(δ+η).
The third estimate in the theorem is an immediate consequence of (12) and the equality
(59)Aηuα-fδ=Aη(uα-u*)+(Aη-A)u*+f-fδ.Before presenting the proof of the main result, we will prove two lemmas: the first regarding the convergence of Lagrange multipliersλα from equality (17) and the second concering the projection of u*-uα on the subspace R(A*A)¯.Lemma 9.
Suppose the conditions (34) and (12) are satisfied and u*∈πU(R(A*), (I-πR(A*)¯)h′(u*)=0. Then, for α=O(δ+η) one has
(60)λαα=O(1).Proof.
From(I-πR(A*)¯)h′(u*)=0, it follows that u*=u∞ and
(61)h′(u*)=πR(A*)¯h′(u*)∈R(A*A).
Then, there is e∈R(A*A) such that h′(u*)=A*Ae and consequently
(62)gα(Aη*Aη)h′(u*)=gα(Aη*Aη)A*Ae=gα(Aη*Aη)(A*A-Aη*Aη)e+(gα(Aη*Aη)Aη*Aη-I)h+h.
Adding and subtracting (I-πη)((A*A-Aη*Aη)/α)e on the right side of this equality and then multiplying both sides by h′(uα), we obtain
(63)〈gα(Aη*Aη)h′(u*),h′(uα)〉=〈(αgα(Aη*Aη)A*A-Aη*Aηαehh-(I-πη)A*A-Aη*Aηαe),h′(uα)〉+〈A*A-Aη*Aηαe,(I-πR(A*)¯)h′(uα)〉+〈(gα(Aη*Aη)Aη*Aη-I)e,h′(uα)〉+〈e,h′(uα)〉.
Observe that ∥(I-πR(A*)¯)((A*A-Aη*Aη)/α)e∥ remains bounded when δ→0, η→0 and α=O(δ+η). Now, using (37), we have that
(64)∥αgα(Aη*Aη)A*A-Aη*Aηαe-(I-πη)A*A-Aη*Aηαe∥=O(∥(I-πη)A*A-Aη*Aηαe∥(δ+η))⟶0.
Besides, (gα(Aη*Aη)Aη*Aη-I)e→0 (Lemma 2) and
(65)(I-πη)h′(uα)⟶(I-πR(A*)¯)h′(u*)⟶0hhhhhhhhhhhhhhasα=O(δ+η)⟶0.
Now, from (63), it follows that
(66)limα→0〈gα(Aη*Aη)h′(u*),h′(uα)〉=〈e,h′(u*)〉=〈e,A*Ae〉=∥Ae∥2≠0.
It means that 〈gα(Aη*Aη)h′(u*),h′(uα)〉≠0 for all sufficiently small δ and η. Multiplying (19) by h′(u*) and taking into account the last inequality, we will obtain
(67)λαα=1α〈h′(u*),vα-uα〉〈gα(Aη*Aη)h′(u*),h′(uα)〉.
If we replace h′(u*) and u* in the previous equality by A*Ae and u∞, it becomes
(68)λαα=1α〈Ae,A(vα-u∞)+A(u*-uα)〉〈gα(Aη*Aη)h′(u*),h′(uα)〉.
Based on Theorem 8, it follows the boundness of ∥A(vα-u∞)∥/α and ∥A(u*-uα)∥/α as α=O(δ+η)→0. Now, it is clear that λα/α also remains bounded as α=O(δ+η).Lemma 10.
Suppose the conditions (34) and (12) are satisfied. If α=O(δ+η), then ∥πR(A*)¯(u*-uα)∥=O(δ+α).Proof.
From(69)u*-uα=u∞-vα+λαgα(Aη*Aη)h′(uα)-γ*(I-πR(A*)¯)h′(u*),
we have that
(70)πη(u*-uα)=πη(u∞-wα)+λαααgα(Aη*Aη)πηh′(uα)-γ*πη(I-πR(A*)¯)h′(u*).
Consequently,
(71)πR(A*)¯(u*-uα)=(πR(A*)¯-πη)×(u*-uα)+πη(u∞-vα)+λαααgα(Aη*Aη)πηh′(uα)-γ*πη(I-πR(A*)¯)h′(u*).
The value λα/α remains bounded as the parameter α=O(δ+η) converges to zero, in case of (I-πR(A*)¯)h′(u*)=0, as well as in case of (I-πR(A*)¯)h′(u*)≠0. Now, for α=O(δ+η), the estimate
(72)∥πR(A*)¯(u*-uα)∥=O(δ+η)
follows from
(73)∥πη-πR(A*)¯∥=O(η),∥u∞-vα∥=O(δ+η),∥πη(I-πR(A*)¯)∥=O(η),∥πηgα(Aη*Aη)∥≤1(mA-η)2.The following theorem is the main result of the paper.Theorem 11.
Suppose the conditions (12) and (34) are satisfied and u*∈πR(A*) and α=O(δ+η). Then,
(74)∥u*-uα∥=O(δ+η).Proof.
From (17), we obtain the following equality:
(75)(I-πR(A*)¯)uα=1α(I-πR(A*)¯)×(A*-Aη*)(fδ-Aηuα)-λαα(I-πR(A*)¯)h′(uα).
From here, taking into account the equality u*=u∞-γ*(I-πR(A*)¯)h′(u*), we have
(76)u*-uα=πR(A*)¯(u*-uα)+(I-πR(A*)¯)u*-(I-πR(A*)¯)uα=πR(A*)¯(u*-uα)+λαα(I-πR(A*)¯)h′(uα)-γ*(I-πR(A*)¯)h′(u*)+1α(I-πR(A*)¯)×(A*-Aη*)(Aηuα-fδ).
Multiplying both sides of this equality by u*-uα and taking into account (13), we obtain
(77)∥u*-uα∥2=〈Aηuα-fδαπR(A*)¯(u*-uα)+(I-πR(A*)¯)hh×(A*-Aη*)Aηuα-fδα,u*-uα〉+γ*〈h′(u*),πR(A*)¯(u*-uα)〉-λαα〈h′(uα),πR(A*)¯(u*-uα)〉.
Now, adding and subtracting (λα/α)〈h′(u*),πR(A*)¯(u*-uα)〉 on the right side of this equality, it becomes
(78)∥u*-uα∥=〈Aηuα-fδαπR(A*)¯(u*-uα)+(I-πR(A*)¯)hhhhhhh×(Aη*-A*)Aηuα-fδα,u*-uα〉+(γ*-λαα)〈h′(u*),πR(A*)¯(u*-uα)〉+λαα〈h′(u*)-h′(uα),πR(A*)¯(u*-uα)〉.
Observe that in case of(I-πR(A*)¯)h′(u*)=0, we can take γ*=0. Then, 0≤〈h′(u*),u*-uα〉=〈h′(u*),πR(A*)¯(u*-uα)〉. From here and (78), we obtain
(79)∥u*-uα∥2≤〈Aηuα-fδαπR(A*)¯(u*-uα)+(I-πR(A*)¯)hhhhhhh×(Aη*-A*)Aηuα-fδα,u*-uα〉+λαα〈h′(u*)-h′(uα),πR(A*)¯(u*-uα)〉.
Finally, from here, by Lemmas 9 and 10, Theorem 8, and conditions (12) and (3), the estimate (74) follows.
In case of(I-πR(A*)¯)h′(u*)≠0, the estimate (74) is an immediate consequence of Lemmas 7 and 9, Theorem 8, and conditions (12).



---

*Source: 506368-2013-11-10.xml*