ITERATIVE SOLUTION OF UNSTABLE VARIATIONAL INEQUALITIES ON APPROXIMATELY GIVEN SETS

The onvergence and the stability of the iterative regularization method for solving variational inequalities with bounded nonsmooth properly monotone (i.e., degenerate) operators in Banach spaces are studied. All the items of the inequality (i.e., the operator A, the “right hand side” f and the set of constraints Ω) are to be perturbed. The connection between the parameters of regularization and perturbations which guarantee strong convergence of approximate solutions is established. In contrast to previous publications by Bruck, Reich and the first author, we do not suppose here that the approximating sequence is a priori bounded. Therefore the present results are new even for operator equations in Hilbert and Banach spaces. Apparently, the iterative processes for problems with perturbed sets of constraints are being considered for the first time.


Introduction and previous results
We consider iterative methods for solving variational inequalities with monotone operators in Banach spaces.A typical example of such problems is the problem of optimization of convex functionals on convex closed sets.Recall that the gradient of a convex functional defined in a Banach space B is a monotone, in general nonlinear and nonsmooth, operator acting from B to the dual space B * .Particular cases of variational inequalities are problems of solving operator equations in Banach spaces, problems of finding saddle points of convex-concave functionals, equations with general elliptic operators, etc. [25,28,17].
Assume that B is a uniformly convex Banach space.Denote by ϕ, x the dual product in B, i.e., the pairing between ϕ ∈ B * and x ∈ B. Let • be a norm in B, and • * be the norm in B * .Let Ω be a convex closed set in B, A be a bounded monotone operator from B to B * with domain D(A) ⊆ B and range R(A) ⊆ B * , Ω ⊆ intD(A) (intD(A) stands for the interior of D(A)).Recall that an operator A is called "bounded" if it maps bounded sets of B onto bounded sets of B * ; an operator A is called "monotone" if for every x 1 , x 2 ∈ B, y 1 ∈ Ax 1 and y 2 ∈ Ax 2 , "strongly monotone" if and "uniformly monotone" if where ψ(t) is a continuous nondecreasing positive function for t > 0, ψ(0) = 0.
If A satisfies some continuity condition (for example, demi-, hemi-continuity) then in the inequalities (1)-(3) we set y 1 = Ax 1 and y 2 = Ax 2 .An operator A is called maximal monotone (in the sense of inclusion) if its graph is a maximal monotone set [19,25,28].
We define A : B → B * to be a "proper monotone" operator if it is monotone and there is no other strengthening of the condition (1) (for example, to the levels of (2) or (3)).
It is well known [1,2] that the problem of finding a solution x * ∈ Ω for the variational inequality with a proper monotone operator A belongs to the class of ill-posed problems.It is unstable with respect to perturbations of all of its terms: the operator A, "the right-hand side" f and the set of constraints Ω.That is why the conventional approximate methods of relaxation type (gradient methods, Newton methods, etc.) do not, in general, converge to the solution x * , which, moreover, does not necessary exist.In compliance with the concept of ill-posed problems here we assume that there exists a nonempty set N of solutions to the variational inequality (4).
By a "solution" of (4) we understand an element x * ∈ Ω for which there exists y ∈ Āx * such that the inequality y − f, x − x * ≥ 0, ∀x ∈ Ω, is satisfied.Here Ā is the maximal monotone extension of A on Ω (which always exists by the Zorn's lemma).If A is continuous, then x * is a classical solution.If A is maximal monotone and Ω = B, then x * is a solution in the sense of inclusion, i.e., f ∈ Ax * .
It is well known that N is a convex closed set (see, for instance, [15]).Due to the reflexivity of B, there exists a unique element x * of minimal norm x * = min x∈N x .(5) Moreover, the concept of an ill-posed problem assumes the application of a regularization procedure to the variational inequality with perturbed initial data, to ensure that the convergence of the approximate solutions is stable.
As regularizing (smoothing) operators we consider the one-parameter family where α is a positive parameter of regularization and J is a normalized duality mapping.Recall that J : B → B * is called the "normalized duality mapping" if The duality mapping exists in every Banach space.The (generally nonlinear) operator J is always monotone, coercive and bounded.It is also hemicontinuous in smooth reflexive Banach space, uniformly continuous on every bounded set in uniformly smooth Banach space, and uniformly monotone on every bounded set in uniformly convex Banach space [9,10].If B is a Hilbert space, then J is the identity operator I.If B is a strictly convex and reflexive along with its dual, and J * : B * → B, is the normalized duality mapping in the space B * , then J * = J −1 , where J −1 is the inverse of the mapping J.If B is smooth, i.e., the norm of B is differentiable in the sense of Gateaux, then Jx = grad( x 2 /2).Let us notice that these same properties are shared by the duality mapping J ν with the gauge function ν(t) : which also plays an important role in the general theory of monotone and accretive operators.
Assume that instead of A, f, and Ω perturbed values A h , f ω and Ω σ , depending on positive parameters h, ω, and σ respectively are given.Furthermore, let Here H X (Q 1 , Q 2 ) is the Hausdorff distance between the sets Q 1 and Q 2 in the space X (see, for example, [2]), M (x) and M h (x) are the ranges of the operators Ā and Āh in x, where Ā and Āh stands for the maximal extensions of the corresponding monotone operators, ζ(t) is continuous non-decreasing function for t ≥ 0, bounded on bounded sets.If A and A h are continuous then (8) is replaced by the inequality In what follows we assume that A h are monotone operators, D(A h ) = D(A) for all h ≥ 0 (A 0 = A), and Ω σ are convex closed sets for all σ ≥ 0, (Ω 0 = Ω), Ω σ ⊂ intD(A).Using S(α, x) = αJx as the smoothing operator we have the following regularized variational inequality: (11) recently studied by Alber and Notik [2,11].
The following two main problems are naturally posed for the variational inequality (11): 1. Find the connection between parameters h, ω, and σ of the problem and the regularization parameter α, securing convergence of the approximate solutions x ∆ α to the solution x * of the initial inequality (4), and their stability.
The simplest of the known relations is the following: ∆/α → 0 as α → 0. Namely, under this very condition convergence x ∆ α → x * is proved in [2,11].However, this choice of α is not single-valued and does not guarantee the regularization properties of the approximate solutions to the variational inequality (11).Therefore, the following problem is stated: 2. Justify a single-valued choice of the regularization parameter α providing regularizing properties of the approximate solutions.In other words, the given perturbing parameters ∆ = (h, ω, σ) have to provide a single-valued choice of α, continuously tending to zero along with ∆ → 0. Then the limit relation x ∆ α → x * is guaranteed.In this case the method of finding solution x ∆ α from (11) along with the way to choose α is called the operator regularization, in contrast to the iterative regularization to be discussed in what follows.
The operator method of regularization has got a great deal of interest.For linear equations with exact data in Hilbert spaces it was studied by Lavrentjev in [27] and Lattes and Lions in [26].Here the regularizing family is especially simple, namely, S(α, x) = αI.
Browder [20] and Bakushinskii [17] used a strongly and uniformly monotone operators M (x) in Hilbert spaces: as the regularizing operators for variational inequalities.Unfortunately, in [20,17] there are no examples of such operators for Banach spaces.In the work [1] Alber has first applied the duality mappings as regularizing operators for solving equations with exact and approximate data in Banach spaces.The duality mappings do not satisfy the condition (12) even in smooth Banach spaces: generally speaking, they are uniformly monotone on bounded sets.This makes apparent difficulties in investigation of the methods mentioned above.Nevertheless, duality mappings turn out to be an exceptionally efficient tool for regularization of unstable problems in Banach spaces (operator equations, variational inequalities, problems for calculating values of unbounded operators, convex optimization problems, etc.).This allowed to get the most of the results about convergence and stability [2,4,11,15,16,24,29,30,35,36] which have been known previously only for Hilbert spaces, mainly, in linear cases (cf.[32]).In addition, let us point out that duality mappings have an analytical descriptions for the most important Banach spaces, namely, l p , L p , H p m , 1 < p < ∞ [3] and Orlicz spaces with Luxemburg norm [12].
It is a special question how to choose the parameter of regularization.The most popular approach is based on the so called the residual principle.For linear equation and regularization of the variational type in Hilbert spaces, it was first proposed and studied by Ivanov and Morozov (see [23,32]).For nonlinear problems and regularization of the operator type in Banach spaces, it was first investigated by Alber in [4] (see also [15]).Let us illustrate the idea of this approach by finding the solution x * to the operator equation Ax = f with properly monotone hemi-continuous operator A acting from a uniformly convex Banach space B to the dual strictly convex space B * .Let, for simplicity, the equation is defined only with a perturbed RHS f ω , ω > 0. According to (11) the regularized equation is In fact [1], x ω α → x * for α → 0 and ω/α → 0. However, the last relations do not provide the regularization properties of (13) in the sense that they do not guarantee convergence x ω α → x * only under the condition ω → 0. In the principle of the residual we choose α = ᾱ from the condition Moreover, x ω ᾱ converge strongly to x * if ω → 0 and 0 < s < 1, and converge weakly if ω → 0 and s = 1.In the both cases as a consequence we get the following limit relations: ᾱ → 0 for ω → 0; ω/ᾱ → 0 for 0 < s < 1, and ω/ᾱ < const for s = 1.This justifies approximating the solution x * by the solutions x ω ᾱ of ( 13) for a fixed ω.In this very form the problem of finding x * ∈ N is mostly close to the reality, and corresponds to the fundamental principle of selection of the approximate solution in correct (stable) problems: residual in the approximate solution has to be satisfied with the accuracy not exceeding the error of the RHS.
Of special importance in applications are equations and variational inequalities with maximal monotone, possibly multi-valued, operators [7,19].Evidently, the residual principle in the form of ( 14) is not applicable in these cases.The reason is that the residual r(ᾱ) becomes a multi-valued function of ᾱ, but in the proof we require this function to be continuous.In order to overcome this difficulty Alber in [4] proposed significant generalization of the residual principle.The idea of it is as follows: on the solutions x ω,h α of the regularized equation with maximal monotone operators A h , the residual is Hence the function r(α) = α x ω,h α is natural to be called the generalized residual on approximate solutions x ω,h α (in other points it is of no interest).The function r(α) is continuous, single-valued and monotone increasing.Moreover, if A is hemi-(demi-)continuous operator then r(α) coincides with the classical residual Ax ω,h α − f ω * .It has been shown in [15] that the generalized residual principle in the form ᾱ is the solution of ( 15) for α = ᾱ is the regularizing algorithm as ω, h → 0. The statement completely corresponds to the case of continuous operator A. The principle of generalized residual proved to be exceptionally fruitful not only for the problems with maximal but even for discontinuous monotone operators.Moreover, Ryazantseva [35] was first to use this principle for variational inequalities where the classical notion of residual is not defined even in the continuous case (see also [16]).The similar principle of the smoothing functional was studied in detail by Alber and Liskovets [8].
At last, let us mention the papers where under consideration were convergence and stability of the regularization method in the operator form for variational inequalities with perturbed sets Ω.In the most complete form it was studied recently by Alber and Notik [11] and Alber [2].Different particular cases appeared earlier in Mosco [31], Bakushinskii [18], Liskovets [30], Ryazantseva [36], etc.The principle of the residual for such problems has been obtained by Ryazantseva [36] for Hilbert spaces, for Banach spaces the problem is still open.
Thus, the operator method of regularizing variational inequalities in Banach spaces has been developed in satisfactory completeness.However, this method by itself does not give a final solution to the problem, -for every fixed α it requires solving a nonlinear problem of type (15).In contrast to this the iterative regularization method gives successive approximations to the solution x * on every step of the iterative process.Here the parameters of the regularization are preassigned.Let us emphasize that the residual principle in the methods of iterative type has been studied in Hilbert spaces for linear equations and for iterative sequences of the form Vainikko [37]).

Main result
The process of proving convergence and stability of iterative regularization includes the operator regularization of the form The method of iterative regularization which we consider can be described by the following iterative procedure: Here xn = P Ω x n is the metric projection of x n on Ω.Since Ω is a convex closed set in a reflexive space then for arbitrary x n ∈ B there exists the unique xn ∈ Ω [3].
Analogously to ( 8) and ( 9) assume that Ω σn ⊂ intD(A) are convex sets, D(A hn ) = D(A) and for x ∈ D(A) the following inequality holds: The method of iterative regularization was first studied by Bruck and Bakushinskii in Hilbert spaces (see [21,17]).They used the following scheme of the proof: according to the inequality on the fist stage convergence to x * of the solutions z n to the variational inequality Az n + α n z n − f, x − z n ≥ 0, ∀x ∈ Ω was established.On the second stage the limit relation lim n→∞ x n − z n = 0, (25) was proved.Here Such a scheme stems from Alber [5] where stabilization of the method of continuous regularization in Hilbert space H for solution of the operator equation Ax = f with conditions

The method of iterative regularization of operator equation
is the difference approximation of the differential equation ( 27) according to the simplest Euler scheme.
In application to variational inequalities the following fundamental statement strikes: a point x * ∈ Ω ⊂ H is a solution of the variational inequality Ax − f, ξ − x ≥ 0, ∀ξ ∈ Ω, if and only if x * is a solution of the operator equation with an arbitrary ε > 0. Successive approximations for the regularized equation (28) with perturbed data give the iterative sequence (26).
In Banach spaces iterative regularization was considered in [6] only for nonlinear equations Ax = f with monotone operators A : B → B * .Transition from Hilbert to Banach spaces is not trivial due to the following reasons.First of all, using conventional Lyapunov functionals * leads to unnatural a priori assumptions being outside the framework of the theory of monotone operators.To avoid this it was proposed in [6] (and earlier in [9]) to use in Banach spaces a new Lyapunov functional that does not require any a priori structure assumptions but monotonicity of the operator A. However, using such functional essentially complicates the proof, and the question of relations between V (Jx, x * ) and the conventional functional V 1 (x) becomes of major importance since the final results should be stated in terms of V 1 (x).
Second, it turned out that in Banach spaces it is necessary to use the geometry of these spaces in order to investigate convergence and stability of iterative processes.In Hilbert spaces there is no such problem, more precisely, the geometry of Hilbert spaces is trivial enough, and it is automatically employed in squaring the norm x − x * in the Lyapunov functional V 1 (x), which is equivalent using the parallelogram equality [22] x + y 2 H + x − y H = 2 x 2 H + 2 y 2 H . Third, in Hilbert spaces the operator (A+αI) of ( 24) is strongly monotone all over the space, in contrast to Banach spaces the operator (A + αJ) is uniformly monotone only on the bounded sets.This is essentially used in the proof of uniform boundedness of approximate iterations and turns out to be the most difficult point in the theory of iterative methods.Now let us summarize the main characteristics which distinct this paper from previous works: 1.In [6] iterative regularization of operator equations in all the space B was studied in the form Jx n+1 = Jx n − ε n (y hn + α n Jx n − f ωn ), y hn ∈ A hn x n , (30) x n = J * Jx n , n = 0, 1, . . ., x 0 ∈ B.
In the present paper we study the iterative regularization method for variational inequalities which uses the operation of metric projection on Ω under the sign of the operator A (cf. [13]) and the additional "penalty" term (19).2. In [6] it was a priori assumed that the sequence (30) is bounded (the same in [21,34]).Here we omit this requirement.Therefore, the results obtained in this paper turn out to be new even for operator equations in Banach spaces.3.In this paper we are first to consider stability of iterative processes for variational inequalities in relation to perturbation of the set Ω.
Denote by δ B (ε), ε ∈ [0, 2] the modulus of convexity, by ρ B (τ ), τ ≥ 0, the modulus of smoothness of B [22], g B (ε) = δ B (ε)/ε.Assume that the inverse function g −1 B (•) exists.It is known that g B (ε) and g −1 B (•) are nondecreasing, and δ B (ε) and ρ B (τ ) are increasing functions, δ B (0) = ρ B (0) = 0. We also consider the function µ(t) = d 1 t 2 + d 2 ρ B (t), which is increasing along with its inverse µ −1 (•) (d 1 and d 2 are positive constants).It is possible to assume that δ B (ε) ≥ cε γ , where γ ≥ 2 and c is a positive constant.This assumption is not too restrictive since, on one hand, a direct calculation of δ B (ε) in the spaces l p , L p , W p m , 1 < p < ∞, and in some Orlicz spaces, shows that it allows the corresponding lower estimate On the other hand, the Pisier theorem [33] states that the same result up to isomorphism is valid for arbitrary uniformly convex Banach space.Thus, without loss of generality, we assume the following: there exists continuous on [0, 2] convex increasing function δB (ε), δB (0) = 0, such that Our aim is to prove the following result.
Theorem 1.Let B be an uniformly convex Banach space, B * be its dual.Moreover, assume that 1. x * is the solution of the variational inequality (4) of minimal norm; 2. The operator A : B → 2 B * is monotone bounded with ϕ-arbitrary growth order, i.e.

The perturbations of the operator A, the element f and the convex set
Ω are given with errors and satisfy (21)-( 23).

Auxiliary lemma
As a preliminary, let us show validity of the following "proximity lemma".
Lemma 1. Suppose that (i) B is a uniformly convex Banach space with the modulus of convexity δ B (ε); (ii) the sequences {z 1 } and {z 2 } of the solutions to the variational inequalities (iii) the operator T 1 is monotone and bounded on the sets {z 1 } and {z 2 }, i.e.
Then the following estimate is valid: Proof.We consider the expression By virtue of monotonicity of T 1 , In [9,10] an estimate of the adjoint mapping was given: Then we have On the other hand, since Analogously, one can obtain the estimate It is evident that And so it follows that From the last inequality and from (36) we have Only two cases are possible, either (i) In the second case, since [22,3] Consider now (37).We have Using (39) , we get Comparing the last expression with (38), we get (35).The proof of the lemma is completed.
Remark 1.If σ 0 = 0, i.e.Ω 1 coincides to Ω 2 and ω = 0, i.e.T 1 coincides with T 2 (at least on the elements of the sequence {z 2 }) then This estimate coincides with those obtained before (see [1,6]).Proximity estimates of solutions to variational inequalities with perturbed operators and perturbed sets of constraints have not exist before.Remark 2. In the case of multi-valued operators, by T 1 and T 2 we mean the single-valued sections of the corresponding operators.

Proof of Theorem 1
Consider the auxiliary inequality (33).Earlier, in [1], it was proved that the relation lim n→∞ z n − x * = 0 holds in B while σ/α → 0 and z n ≤ M 1 , M 1 = const.From (24) it follows that it is necessary to prove only (25) to obtain strong convergence and stability of the method ( 18)- (19).
Let z m and z m+1 be solutions of the variational inequalities where x and z are arbitrary points from B. Let us state the properties of this functional (see [3]): 1. V (Jx, z) is a convex, continuous and differentiable with respect to each argument ϕ = Jx and z; From the convexity of V (Jx, z) in z we have Now from the convexity of V (Jx, z) in ϕ = Jx, we have (42) (41) together with (42) give the following inequality Let us estimate each of the four terms in the RHS of (43).
The first term.Using the estimate for the modulus of continuity of the duality mapping [9,10], we get , where Then, from Lemma 1 it follows that We have used here the inequality: The second term.First consider the following estimate ( see [9,10]) , where It is evident (see e.g. ( 22)), that f ωn * ≤ f * + ω n .By virtue of ( 21) and (32) Therefore Taking into account that where And, finally, the fourth term.
Now we can rewrite (43) in the following form: and If x n ≤ K 1 and xn ≤ K 2 , then it is possible to derive from the expressions (44), ( 45), ( 46), ( 47), (49) the following inequality: where c6 , c7 , c8 , c9 , c10 are absolute constants.Getting back to (48), we obtain It is evident that Thus, from (57), it follows that Here, Taking all this into consideration, along with (55), we have the following numerical inequality: Recall that m = n + n 0 .Choose n 0 according to the formula: Let us calculate the Lyapunov functional V (Jx, z) at the point (x n+1 , z m+1 ).
Let us estimate the last two terms.
Remark 4. In the proof of Theorem 1 the monotonicity of the operators A hn is used only in (8), to provide the existence of the maximal monotone extension.Thus, continuous A hn could be not necessary monotone.

Remark 3 .
This scheme of the proof could be applied to the study of equations Ax = f with accretive operators A : B → B :J(x − y), Ax − Ay ≥ 0, ∀x, y ∈ B.It gives the possibility to get rid of the assumption of the a priori boundedness of the sequence {x n } in the method of iterative regularizationx n+1 = x n − ε m (y hm x n + α m x n − f ωm ), n = 1, 2, ...
we can write for all y n ∈ Ax n y hn * ≤ y hn − y n * + y n * ≤ h n ζ( x n ) + ϕ( x n ).Now let us estimate Jx n+1 − Jx n * :