Regularization of ill-posed mixed variational inequalities with non-monotone perturbations

Abstract

In this paper, we study a regularization method for ill-posed mixed variational inequalities with non-monotone perturbations in Banach spaces. The convergence and convergence rates of regularized solutions are established by using a priori and a posteriori regularization parameter choice that is based upon the generalized discrepancy principle.

1 Introduction

Variational inequality problems in finite-dimensional and infinite-dimensional spaces appear in many fields of applied mathematics such as convex programming, nonlinear equations, equilibrium models in economics, and engineering (see [1–3]). Therefore, methods for solving variational inequalities and related problems have wide applicability. In this paper, we consider the mixed variational inequality: for a given f ∈ X*, find an element x₀ ∈ X such that

(1)

where A : X → X* is a monotone-bounded hemicontinuous operator with domain D(A) = X, φ : X → ℝ is a proper convex lower semicontinuous functional and X is a real reflexive Banach space with its dual space X*. For the sake of simplicity, the norms of X and X* are denoted by the same symbol || · ||. We write 〈x*, x〉 instead of x*(x) for x* ∈ X* and x ∈ X.

By S₀ we denote the solution set of the problem (1). It is easy to see that S₀ is closed and convex whenever it is not empty. For the existence of a solution to (1), we have the following well-known result (see [4]):

Theorem 1.1. If there exists u ∈ dom φ satisfying the coercive condition

(2)

then (1) has at least one solution.

Many standard extremal problems can be considered as special cases of (1). Denote φ by the indicator function of a closed convex set K in X,

Then, the problem (1) is equivalent to that of finding x₀ ∈ K such that

(3)

In the case K is the whole space X, the later variational inequality is of the form of the following operator equation:

(4)

When A is the Gâteaux derivative of a finite-valued convex function F defined on X, the problem (1) becomes the nondifferentiable convex optimization problem (see [4]):

(5)

Some methods have been proposed for solving problem (1), for example, the proximal point method (see [5]), and the auxiliary subproblem principle (see [6]). However, the problem (1) is in general ill-posed, as its solutions do not depend continuously on the data (A, f, φ), we used stable methods for solving it. A widely used and efficient method is the regularization method introduced by Liskovets [7] using the perturbative mixed variational inequality:

(6)

where A _h is a monotone operator, α is a regularization parameter, U is the duality mapping of X, x _* ∈ X and (A _h , f _δ , φ _ε ) are approximations of (A, f, φ), τ = (h, δ, ε). The convergence rates of the regularized solutions defined by (6) are considered by Buong and Thuy [8].

In this paper, we do not require A _h : to be monotone. In this case, the regularized variational inequality (6) may be unsolvable. In order to avoid this fact, we introduce the regularized problem of finding such that

(7)

where μ is positive small enough, U^s is the generalized duality mapping of X (see Definition 1.3) and is in X which plays the role of a criterion of selection, g is defined below.

Assume that the solution set S₀ of the inequality (1) is non-empty, and its data A, f, φ are given by A _h , f _δ , φ _ε satisfying the conditions:

(1) || f - f _δ || ≤ δ, δ → 0;

(2) A _h : X → X* is not necessarily monotone, D(A _h ) = D(A) = X, and

(8)

with a non-negative function g(t) satisfying the condition

(3) φ _ε : X → ℝ is a proper convex lower semicontinuous functional for which there exist positive numbers c _ε and r _ε such that

and

(9)

(10)

where C₀ is some positive constant, d(t) has the same properties as g(t).

In the next section we consider the existence and uniqueness of solutions of (7), for every α > 0. Then, we show that the regularized solutions converge to x₀ ∈ S₀, the -minimal norm solution defined by

The convergence rate of the regularized solutions to x₀ will be established under the condition of inverse-strongly monotonicity for A and the regularization parameter choice based on the generalized discrepancy principle.

We now recall some known definitions (see [9–11]).

Definition 1.1. An operator A : D(A) = X → X* is said to be

• (a) hemicontinuous if A(x + t _n y) ⇀ Ax as t _n → 0⁺, x, y ∈ X, and demicontinuous if x _n → x implies Ax _n ⇀ Ax;

• (b) monotone if 〈Ax - Ay, x - y〉 ≥ 0, ∀x, y ∈ X;

• (c) inverse-strongly monotone if

  

  (11)

where m _A is a positive constant.

It is well-known that a monotone and hemicontinuous operator is demicontinuous and a convex and lower semicontinuous functional is weakly lower semicontinuous (see [9]). And an inverse-strongly monotone operator is not strongly monotone (see [10]).

Definition 1.2. It is said that an operator A : X → X* has S-property if the weak convergence x _n ⇀ x and 〈Ax _n - Ax, x _n - x〉 → 0 imply the strong convergence x _n → x as n → ∞.

Definition 1.3. The operator U^s : X → X* is called the generalized duality mapping of X if

(12)

When s = 2, we have the duality mapping U. If X and X* are strictly convex spaces, U^s is single-valued, strictly monotone, coercive, and demicontinuous (see [9]).

Let X = L^p (Ω) with p ∈ (1, ∞) and Ω ⊂ ℝ ^m measurable, we have

Assume that the generalized duality mapping U^s satisfies the following condition:

(13)

where m _s is a positive constant. It is well-known that when X is a Hilbert space, then U^s = I, s = 2 and m _s = 1, where I denotes the identity operator in the setting space (see [12]).

2 Main result

Lemma 2.1. Let X* be a strictly convex Banach space. Assume that A is a monotone-bounded hemicontinuous operator with D(A) = X and conditions (2) and (3) are satisfied. Then, the inequality (7) has a non-empty solution set S _ε for each α > 0 and f _δ ∈ X*.

Proof. Let x _ε ∈ dom φ _ε . The monotonicity of A and assumption (3) imply the following inequality:

for ||x|| > r _ε . Consequently, (2) is fulfilled for the pair (A + αU^s , φ _ε ). Thus, for each α > 0 and f _δ ∈ X*, there exists a solution of the following inequality:

(14)

Observe that the unique solvability of this inequality follows from the monotonicity of A and the strict monotonicity of U^s . Indeed, let x₁ and x₂ be two different solutions of (14). Then,

(15)

and

(16)

Putting z = x₂ in (15) and z = x₁ in (16) and add the obtained inequalities, we obtain

Due to the monotonicity of A and the strict monotonicity of U^s , the last inequality occurs only if x₁ = x₂.

Let be a solution of (14), that is,

(17)

For all h > 0, making use of (8), from (17) one gets

(18)

Since μ ≥ h, we can conclude that each is a solution of (7).

□

Let be a solution of (7). We have the following result.

Theorem 2.1. Let X and X* be strictly convex Banach spaces and A be a monotone-bounded hemicontinuous operator with D(A) = X. Assume that conditions (1)-(3) are satisfied, the operator U^s satisfies condition (13) and, in addition, the operator A has the S-property. Let

(19)

Thenconverges strongly to the-minimal norm solution x₀ ∈ S₀.

Proof. By (1) and (7), we obtain

This inequality is equivalent to the following

(20)

The monotonicity of A, assumption (1), and the inequalities (8), (9), (13) and (20) yield the relation

(21)

Since μ/α → 0 as α → 0 (and consequently, h/α → 0), it follows from (19) and the last inequality that the set are bounded. Therefore, there exists a subsequence of which we denote by the same weakly converges to .

We now prove the strong convergence of to . The monotonicity of A and U^s implies that

(22)

In view of the weak convergence of to , we have

(23)

By virtue of (8),

(24)

Using further (7), we deduce

(25)

Since and φ _ε is proper convex weakly lower semicontinuous, we have from (25) that

(26)

By (22)-(24) and (26), it results that

Finally, the S property of A implies the strong convergence of to .

We show that . By (8) and take into account (7) we obtain

(27)

Since the functional φ is weakly lower semicontinuous,

(28)

Since is bounded, by (9), there exists a positive constant c₂ such that

(29)

By letting α → 0 in the inequality (7), provided that A is demicontinuous, from (8), (9), (28), (29) and condition (1) imply that

This means that .

We show that . Applying the monotonicity of U^s and the inequalities (8), (9) and (13), we can rewrite (17) as

Since α → 0, ε/α, δ/α, μ/α → 0 (and h/α → 0), the last inequality becomes

Replacing x by , t ∈ (0, 1) in the last inequality, dividing by (1 - t) and then letting t to 1, we get

or

Using the property of U^s , we have that , ∀x ∈ S₀. Because of the convexity and the closedness of S₀, and the strictly convexity of X, we can conclude that . The proof is complete.

□

Now, we consider the problem of choosing posteriori regularization parameter such that

To solve this problem, we use the function for selecting by generalized discrepancy principle, i.e. the relation is constructed on the basis of the following equation:

(30)

with , where is the solution of (7) with , c is some positive constant.

Lemma 2.2. Let X and X* be strictly convex Banach spaces and A : X → X* be a monotone-bounded hemicontinuous operator with D(A) = X. Assume that conditions (1), (2) are satisfied, the operator U^s satisfies condition (13). Then, the function is single-valued and continuous for α≥ α₀> 0, whereis the solution of (7).

Proof. Single-valued solvability of the inequality (7) implies the continuity property of the function ρ(α). Let α₁, α₂ ≥ α₀ be arbitrary (α₀> 0). It follows from (7) that

(31)

where and are solutions of (7) with α = α₁ and α = α₂. Using the condition (2) and the monotonicity of A, we have

It follows from (13) and the last inequality that

Obviously, as μ → 0 and α₁ → α₂. It means that the function is continuous on [α₀; +∞). Therefore, ρ(α) is also continuous on [α₀; +∞).

Theorem 2.2. Let X and X* be strictly convex Banach spaces and A : X → X* be a monotone-bounded hemicontinuous operator with D(A) = X. Assume that conditions (1)-(3) are satisfied, the operator U^s satisfies condition (13). Then

(i) there exists at least a solutionof the equation (30),

(ii) let μ, δ, ε → 0. Then

(1);

(2) if 0 < p < q then, with-minimal norm and there exist constants C₁, C₂> 0 such that for sufficiently small μ, δ, ε > 0 the relation

(32)

holds.

Proof.

(i) For 0 < α < 1, it follows from (7) that

Hence,

We invoke the condition (1), the monotonicity of A, (8), (10), (12), and the last inequality to deduce that

(33)

It follows from (33) and the form of ρ(α) that

Therefore, lim_α→+0α^qρ(α) = 0.

On the other hand,

Since ρ(α) is continuous, there exists at leat one which satisfies (30).

(ii) It follows from (30) and the form of that

Therefore, as μ, δ, ε → 0.

If 0 < p < q, it follows from (30) and (32) that

So,

By Theorem 2.1 the sequence converges to x₀ ∈ S₀ with -minimal norm as μ, δ, ε → 0.

Clearly,

therefore, there exists a positive constant C₂ such that (32). On the other hand, because c > 0 so there exists a positive constant C₁ satisfied (32). This finishes the proof.

□

Theorem 2.3. Let X be a strictly convex Banach space and A be a monotone-bounded hemicontinuous operator with D(A) = X. Suppose that

(i) for each h, δ, ε > 0 conditions (1)-(3) are satisfied;

(ii) U^s satisfies condition (13);

(iii) A is an inverse-strongly monotone operator from X into X*, Fréchet differentiable at some neighborhood of x₀ ∈ S₀and satisfies

(34)

(iv) there exists z ∈ X such that

then, if the parameter α = α (μ, δ, ε) is chosen by (30) with 0 < p < q, we have

Proof. By an argument analogous to that used for the proof of the first part of Theorem 2.1, we have (21). The boundedness of the sequence follows from (21) and the properties of g(t), d(t) and α. On the other hand, based on (20), the property of U^s and the inverse-strongly monotone property of A we get that

Hence,

Further, by virtue of conditions (iii), (iv) and the last estimate, we obtain

Consequently, (21) has the form

(35)

When α is chosen by (30), it follows from Theorem 2.1 that

and

Therefore, it follows from (35) that

where , i = 1, 2, 3, are the positive constants. Using the implication

we obtain

Remark 2.1 If α is chosen a priori such that α ~ (μ + δ + ε) ^η , 0 < η < 1, it follows from (35) that

Therefore,

Remark 2.2 Condition (34) was proposed in [13] for studying convergence analysis of the Landweber iteration method for a class of nonlinear operators. This condition is used to estimate convergence rates of regularized solutions of ill-posed variational inequalities in [14].

Remark 2.3 The generalized discrepancy principle for regularization parameter choice is presented in [15] for the ill-posed operator equation (4) when A is a linear and bounded operator in Hilbert space. It is considered and applied to estimating convergence rates of the regularized solution for equation (4) involving an accretive operator in [16].