Self-adaptive subgradient extragradient-type methods for solving variational inequalities

In this paper, we introduce two subgradient extragradient-type algorithms for solving variational inequality problems in the real Hilbert space. The first one can be applied when the mapping f is strongly pseudomonotone (not monotone) and Lipschitz continuous. The first algorithm only needs two projections, where the first projection onto closed convex set C and the second projection onto a half-space Ck\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{k}$\end{document}. The strong convergence theorem is also established. The second algorithm is relaxed and self-adaptive; that is, at each iteration, calculating two projections onto some half-spaces and the step size can be selected in some adaptive ways. Under the assumption that f is monotone and Lipschitz continuous, a weak convergence theorem is provided. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.


Introduction
Let H be a real Hilbert space with the inner product ·, · . The classic variational inequality problems for f on C are to find a point x * ∈ C such that f x * , xx * ≥ 0, ∀x ∈ C, (1.1) where C is a nonempty closed convex subset of H, and f is a mapping from H to H. The problem and its solution set will be denoted by VI(C, f ) and SOL(C, f ). This class of variational inequality problems arises in many fields such as optimal control, optimization, partial differential equations, and some other nonlinear problems; see [1] and the references therein. Nowadays, variational inequality problems with uncertain data are a very interesting topic, and the robust optimization has recently emerged as a powerful approach to deal with mathematical programming problems with data uncertainty. For more details, we refer the readers to [18,29,30]. In order to solve variational inequality problems, many solution methods have been introduced [7,12,14,16,17,31]. Among these methods, subgradient extragrdeint method has attracted much attention. This method has the following form: (1.2) x k+1 = P T k x kτ f y k , (1.3) where T k := {w ∈ H| x kτ f (x k )y k , wy k ≤ 0} is a half-space and τ > 0 is a constant.
Under the assumption that f is monotone and L-Lipschitz continuous, the method (1.2)-(1.3) weakly converges to a solution to VI(C, f ). The subgradient extragradient method has received a great deal of attention and many authors modified and improved it in various ways, see [9,10,25,35]. To the best of our knowledge, almost subgradient extragradient-type algorithms about variational inequality problems need to assume f is monotone Lipschitz continuous and need one projection on C. These observations lead us to the following concerns:

Question 2 Can we propose a new subgradient extragradient-type projection algorithm such that the projection on C can be replaced by half-space?
In this paper, the main purpose is to solve the two questions. We first introduce a subgradient extragradient-type algorithm for solving Question 1. Under the assumptions that f is strongly pseudomonotone(not monotone) and Lipschitz continuous, we establish the strong convergence theorem. The first algorithm has the following form: where C k is a half-space (a precise definition will be given in Sect. 3 (3.2)). The main feature of the new algorithm is that: it only requires the strong pseudomonotonicity (not monotone) and Lipschitz continuity (there is no need to know or estimate the Lipschitz constant of f ) of the involving mapping instead of the monotonicity and L-Lipschitz continuity conditions as in [4]. Moreover, in the new algorithm, the step sizes λ k do not necessarily converge to zero, and we get the strong convergence theorem. Note that, in the algorithms (1.2)-(1.3) and (1.4)-(1.5), one still needs to execute one projection onto the closed convex set C at each iteration. If C has a simple structure (e.g., half-space, a ball, or a subspace), the projection P C can be computed easily. But, if C is a general closed convex set, one has to solve the minimal distance problem to compute the projection onto C, which is complicated in general.
To overcome this flaw, we present the second algorithm, named two-subgradient extragradient algorithm, for solving monotone and Lipschitz continuous variational inequality problems defined on a level set of a convex function. The two-subgradient extragradient algorithm has the following form: where C k is a half-space (a precise definition will be given in Sect. 4). It is well known that the projection onto half-space can be calculated directly. Clearly, the two-subgradient extragradient algorithm is easy to implement. We prove that the sequence generated by the algorithm (1.6)-(1.7) weakly converges to a solution to VI(C, f ) for the case where the closed convex set C can be represented as a lower level set of a continuously differentiable convex function. Moreover, the step size λ k in this algorithm can be selected in some adaptive way; that is, we have no need to know or to estimate any information as regards the Lipschitz constant of f . Our paper is organized as follows: In Sect. 2, we collect some basic definitions and preliminary results. In Sect. 3, we propose an extragradient-type algorithm and analyze its convergence and convergence rate. In Sect. 4, we consider the two-subgradient extragradient algorithm and analyze its convergence and convergence rate. Numerical results are reported in Sect. 5.

Preliminaries
In this section, we recall some definitions and results for further use. For the given nonempty closed convex set C in H, the orthogonal projection from H to C is defined by (2.1) We write x k → x and x k x to indicate that the sequence {x k } converges strongly and weakly to x, respectively.
Remark 2.1 We claim that property (b) guarantees that VI(C, f ) have one solution at most.
Adding these two inequalities yields σ vu 2 ≤ 0, which implies u = v. Note also that property (b) and the continuity of f guarantee that VI(C, f ) has a unique solution [19].
In the following, we introduce an example to illustrate that f is strongly pseudomonotone but is not monotone in general.
Example 2.1 Let H = l 2 be a real Hilbert space whose elements are square-summable sequences of real scalars, i.e., Let α and β be two positive real numbers such that β < α and 1αβ < 0. Let us set We now show that f is strongly pseudomonotone.
where c (x) is called the Gâteaux differential of c at x.

Definition 2.5 For a convex function c : H → R, c is said to be subdifferentiable at a point
for all sufficiently large k.
Note that Q-linear convergence rate implies R-linear convergence rate; see [ [23], Sect. 9.3]. We remark here that R-linear convergence does not imply Q-linear convergence in general. We consider one simple example, which is derived from [15].
Example 2.2 Let {x k } ∈ R be the sequence of real numbers defined by {x k } converges to 0 with an R-linear convergence rate. Note that this implies {x k } does not converge to 0 with the Q-linear convergence rate.
The following well-known properties of the projection operator will be used in this paper.

Lemma 2.2 ([2]) Let f : H → (-∞, +∞] be convex. Then the following are equivalent:
(a) f is weakly sequential lower semicontinuous; (b) f is lower semicontinuous. (Opial)) Assume that C is a nonempty subset of H, and {x k } is a sequence in H such that the following two conditions hold: (ii) every sequential weak cluster point of {x k } belongs to C. Then {x k } converges weakly to a point in C.
Remark 2.2 According to Sect. 5: Application in [13], we remark here that determining η is an easy and/or feasible task.

Convergence of subgradient extragradient-type algorithm
In this section, we introduce a self-adaptive subgradient extragradient-type method for solving variational inequality problems. The nonempty closed convex set C will be given as follows: We define the half-space as In order to prove our theorem, we assume that the following conditions are satisfied: C1: The mapping f : H → H is σ -strongly pseudomonotone and Lipschitz continuous (but we have no need to know or estimate the Lipschitz constant of f ). Note that strong pseudomonotonicity and the continuity of f guarantee that VI(C, f ) has a unique solution denoted by x * . Now we propose our Algorithm 1.
The following lemma gives an explicit formula of P C k .

Lemma 3.2
The sequence {λ k } is nonincreasing and is bounded away from zero. Moreover, there exists a number m > 0 such that Proof Since δ ∈ (0, 1), it is easy to see the sequence {λ k } is nonincreasing. We claim this sequence is bounded away from zero. Suppose, on the contrary, that λ k → 0. Then, there exists a subsequence {λ k i } ⊂ {λ k } such that Let L be the Lipschitz constant of f , we have Obviously, this inequality contradicts the fact λ k → 0. Therefore, there exists a number m > 0 such that for all k ≥ m. Since the sequence {λ k } is nonincreasing, the preceding relation (3.3) implies λ k ≥ λ m ≥ λ -1 δ m+1 for all k.
The following lemma plays a key role in our convergence analysis. There exists a number m > 0 such that Noting that y k = P C (x kλ k f (x k , v k )), this implies that or equivalently By (3.4), (3.6), and Lemma 3.2, ∀k ≥ m, we get where the third term in the right-hand side of (3.7) is estimated by the strong pseudomonotonicity of f .

Theorem 3.1 Suppose that condition C1 is satisfied. Then the sequences {x k } and {y k } generated by Algorithm 1 strongly converge to a unique solution to VI(C, f ).
Proof Let x * ∈ SOL(C, f ). Using Lemma 3.3, there exists a number m > 0 such that for all k ≥ m, or equivalently Continuing, we get for all integers n ≥ 0, Since the sequence { n k=0 (1ρ 2 ) x ky k 2 + n k=0 2σ λ k y kx * 2 } is monotonically increasing and bounded, we obtain That is, x k → x * and y k → x * . This completes the proof.
We note that Algorithm 1 can give convergence when f is strongly pseudomonotone and Lipschitz continuous without P C k in Step 3. We now give a convergence result via the following new method.

Theorem 3.2 Assume that condition C1 is satisfied. Let
(3.14) Then the sequences {x k } and {y k } generated by (3.14) strongly converge to a unique solution to VI(C, f ).
Proof Similar to the proof of Theorem 3.1, it is not difficult to get a conclusion. We omit the proof here.
Before ending this section, we provide a result on the linear convergence rate of the iterative sequence generated by Algorithm 1.

Theorem 3.3 Let f : C → H be strongly pseudomonotone and L-Lipschitz continuous mapping. Then the sequence generated by Algorithm 1 converges in norm to the unique solution x * of VI(C, f ) with a Q-linear convergence rate.
Proof It follows from Lemma 3.3 that By Lemma 3.2(2), we have Note that Thus, we get Hence, (3.15) where μ := √ 1β ∈ (0, 1). The inequality (3.15) shows that {x k } converges in norm to x * with a Q-linear convergence rate.

Convergence of two-subgradient extragradient algorithm
In this section, we introduce an algorithm named two-subgradient extragradient algorithm, which replaces the first projection in Algorithm 1 onto closed convex set C with a projection onto a specific constructible half-space C k . We assume that C is the same form given in (3.1). In order to prove our theorem, we assume the following conditions are satisfied: The following lemma plays an important role in our convergence analysis.

Lemma 4.1
For any x * ∈ SOL(X, F) and let the sequences {x k } and {y k } be generated by Algorithm 2. Then we have for all k ≥ m.
Proof By an argument very similar to the proof of Lemma 3.3, it is not difficult to get the following inequality: where the second inequality follows by the monotonicity of f . The subsequent proof is divided into the following two cases: Case 1: f (x * ) = 0, then Lemma 4.1 holds immediately in view of (4.1).
Case 2: f (x * ) = 0. By Lemma 2.4, we have x * ∈ ∂C and there exists η > 0 such that f (x * ) = -ηc (x * ). Because c(·) is differentiable convex function, it follows Note that c(x * ) = 0 due to x * ∈ ∂C, we have Since y k ∈ C k and by the definition of C k in step 3, we have By the convexity of c(·), we have c y k + c y k , x ky k ≤ c x k .
Proof Let x * ∈ SOL(C, f ). Using Lemma 4.1, there exists a number m > 0 such that for all k ≥ m. By Remark 2.2, determining η is an easy and/or feasible task. So, find a number λ -1 ≤ 1-ρ 2 2ηK is a feasible task. Since {λ k } is nonincreasing, we get λ k ≤ 1-ρ 2 2ηK for all k ≥ 0. Thus, we have 1ρ 2 -2λ k ηK ≥ 0, which implies that lim k→∞ x kx * exists and lim k→∞ x ky k = 0. Thus, the sequence {x k } is bounded. Consequently, {y k } is also bounded. Letx ∈ H be a sequential weak cluster of {x k }, then there exists a subsequence {x k i } of {x k } such that lim i→∞ x k i =x. Since x ky k → 0, we also have lim i→∞ y k i =x. Due to y k i ∈ C k i and the definition of C k i , Since c (·) is Lipschitz continuous and {x k i } is bounded, we deduce that c (x k i ) is bounded on any bounded sets of H. This fact means that there exists a constant M > 0 such that Because c(·) is convex and lower semicontinuous, using Lemma 2.2, we get c(·) is weak sequential lower semicontinuous. Thus, by combining (4.5) and Definition 2.3, we obtain which meansx ∈ C. Now we turn to showx ∈ SOL(C, f ).
By Lemma 3.2, we get λ k > 0 is bounded away from zero. Passing to the limit in (4.7), we have By Lemma 2.5, we havex ∈ SOL(C, f ). Therefore, we proved that (1) lim k→∞ x kx * exists; (2) If x k i x thenx ∈ SOL(C, f ).
It follows from Lemma 2.3 that the sequence {x k } converges weakly to a solution to VI(C, f ).
Before ending this section, we prove the convergence rate of the iterative sequence generated by Algorithm 2 in the ergodic sense. The base of the complexity proof ( [8]) is In order to prove the convergence rate, the following key inequality is needed. Indeed, by an argument very similar to the proof of Lemma 4.1, it is not difficult to get the following result.
Lemma 4.2 Let {x k } and {y k } be two sequences generated by Algorithm 2, and let λ k be selected as Step 1 in Algorithm 2. Suppose conditions A1-A3 are satisfied. Then for any u ∈ C, we get Theorem 4.2 For any integer n > 0, there exists a sequence {z n } satisfying z n x * ∈ SOL(C, f ) and where z n = n k=0 2λ k y k T n and T n = n k=0 2λ k . (4.8) Proof By Lemma 4.2, we have Summing (4.9) over k = 0, 1, . . . , n, we have (4.10) Combining (4.8) and (4.10), we derive Note that from the fact that y k x * ∈ SOL(C, f ) and z n is a convex combination of y 0 , y 1 , . . . , y k , we get z n x * ∈ SOL(C, f ). This completes the proof.
The preceding inequality implies Algorithm 2 has O( 1 n ) convergence rate. For a given accuracy > 0 and any bounded subset X ⊂ C, Algorithm 2 achieves f (u), z nu ≤ in at most r 2α iterations, where r = sup{ x 0u 2 |u ∈ X}.

Numerical experiments
In order to evaluate the performance of the proposed algorithms, we present numerical examples relative to the variational inequalities. In this section, we provide some numerical experiments to demonstrate the convergence of our algorithms and compare the algorithms we proposed with the existing algorithms in [11,21,28] In this case, we can verify that x * = 0 is the solution to VI(C, f ). We note that f is 1strongly pseudomonotone and 4-Lipschitz continuous on R n (See Example 3.3 [20]) and is not (strongly) monotone. It means that when the methods in [11,28] are applied to solve Example 5.1, its iteration point sequence may not converge to the solution point.

Figure 3
Comparison of the number of iterations of Algorithm 1 and Algorithm 2 with algorithms in [11,21,28] for Example 5.3 [28]. We choose x 0 = x T , x -1 = x T x T , y -1 = (0, 0, . . . , 0) T as the initial points for Algorithm 1 and Algorithm 2; x 1 = x T , x 0 = x T x T as the initial points for [21,28]; x 0 = x T as the initial point for [11], where x T = (1, 1, . . . , 1) T . The numerical results of Example 5.3 are shown in Fig. 3, where it can be seen that the behavior of Algorithm 1 and Algorithm 2 is better than the algorithms in [11,21,28].