On the Weak Convergence of the Extragradient Method for Solving PseudoMonotone Variational Inequalities
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Abstract
In infinitedimensional Hilbert spaces, we prove that the iterative sequence generated by the extragradient method for solving pseudomonotone variational inequalities converges weakly to a solution. A class of pseudomonotone variational inequalities is considered to illustrate the convergent behavior. The result obtained in this note extends some recent results in the literature; especially, it gives a positive answer to a question raised in Khanh (Acta Math Vietnam 41:251–263, 2016).
Keywords
Variational inequality Extragradient method Pseudomonotonicity Weak convergenceMathematics Subject Classification
47J20 49J40 49M301 Introduction
Variational inequalities serve as a powerful mathematical model, which unifies important concepts in applied mathematics like systems of nonlinear equations, necessary optimality conditions for optimization problems, complementarity problems, obstacle problems, or network equilibrium problems [1]. Therefore, this model has numerous applications in the fields of engineering, mathematical programming, network economics, transportation research, game theory, and regional sciences [2].
Several techniques for the solution of a variational inequality (VI) in finitedimensional spaces have been suggested such as projection method, extragradient method, Tikhonov regularization method and proximal point method; see, e.g., [1]. Typically, for guaranteeing the convergence to a solution of the VI, some kinds of monotonicity of the assigned mapping is required. In case of gradient maps, generalized monotonicity characterizes generalized convexity of the underlying function [3]. The wellknown gradient projection method can be successfully applied for solving strongly monotone VIs and inverse strongly monotone VIs [1, 4]. In practice, these assumptions are rather strong. The Tikhonov regularization and proximal point methods can serve as an efficient solution method for solving monotone VIs. For pseudomonotone VIs, however, it may happen that every regularized problem generated by the Tikhonov regularization (resp. every problem generated by the proximal point method) is not pseudomonotone [5]. This implies that the regularization procedures performed in Tikhonov regularization and proximal point methods may destroy completely the given pseudomonotone structure of the original problem and can make auxiliary problems more difficult to solve than the original one.
To overcome this drawback, Korpelevich introduced the extragradient method [6]. In the original paper, this method was applied for solving monotone VIs in finitedimensional spaces. It is a known fact [1, Theorem 12.2.11] that the extragradient method can be successfully applied for solving pseudomonotone VIs. Because of its importance, extragradienttype methods have been widely studied and generalized [1].
Recently, the extragradient method has been considered for solving VIs in infinitedimensional Hilbert spaces [7, 8, 9]. Providing that the VI has solutions and the assigned mapping is monotone and Lipschitz continuous, it is proved that the iterative sequence generated by the extragradient method converges weakly to a solution. However, as stated in [9, Section 6, Q2], it is not clear if the weak convergence is still available when monotonicity is replaced by pseudomonotonicity. The aim of this paper is to give a positive answer to this question. As a consequence, the scope of the related optimization problems can be enlarged from convex optimization problems to pseudoconvex optimization problems. This guarantees the advantage of extragradient method in comparing with the other solution methods.
The paper is organized as follows: We first recall some basic definitions and results in Sect. 2. The weak convergence of the extragradient method for solving pseudomonotone, Lipschitz continuous VIs is discussed in Sect. 3. An example is presented in Sect. 4 to illustrate the behavior of the extragradient method. We conclude the note with some final remarks in Sect. 5.
2 Preliminaries
Remark 2.1
\(u^*\in \mathrm{Sol}(K, F)\) if and only if \(u^*=P_K(u^*\lambda F(u^*))\) for all \(\lambda >0\).
We recall some concepts which are useful in the sequel.
Definition 2.1
 (a)pseudomonotone if$$\begin{aligned} \langle F(u),vu\rangle \ge 0\;\Rightarrow \; \langle F(v),vu\rangle \ge 0 \quad \forall u,v \in H; \end{aligned}$$
 (b)monotone if$$\begin{aligned} \langle F(u)F(v), uv\rangle \ge 0\quad \forall u,v \in H; \end{aligned}$$
 (c)Lipschitz continuous if there exists \(L>0\) such that$$\begin{aligned} \Vert F(u)F(v)\Vert \le L \Vert uv \Vert \quad \forall u,v \in H; \end{aligned}$$
 (d)
sequentially weakly continuous if for each sequence \(\{u^n\}\) we have: \(\{u^n\}\) converges weakly to u implies \(\{F(u^n)\}\) converges weakly to F(u).
Remark 2.2
It is clear that monotonicity implies pseudomonotonicity. However, the converse does not hold. For example, the mapping \(F:\left]0,+\infty \right[ \rightarrow \left]0,+\infty \right[\), defined by \(F(u)=\frac{a}{a+u}\) with \(a>0\) is pseudomonotone but not monotone.
We recall a result which is called Minty lemma [11, Lemma 2.1].
Proposition 2.1
3 Weak Convergence of the Extragradient Method
In this section, we consider the problem VI(K, F) with K being nonempty, closed, convex and F being pseudomonotone on H and Lipschitz continuous with modulus \(L>0\) on K. We also assume that the solution set Sol(K, F) is nonempty.

Data: \(u^0\in K\) and \(\{ \lambda _k \} \in [a,b]\), where \(0<a\le b<1/L\).

Step 0: Set \(k=0\).

Step 1: If \(u^k=P_K(u^k\lambda _k F(u^k))\) then stop.
 Step 2: Otherwise, set$$\begin{aligned} \bar{u}^k= & {} P_K(u^k\lambda _k F(u^k)),\\ u^{k+1}= & {} P_K(u^k\lambda _k F(\bar{u}^k)). \end{aligned}$$
Remark 3.1
If at some iteration we have \(F(u^k)=0\), then \(u^k=P_K(u^k\lambda _k F(u^k))\) and the Extragradient Algorithm terminates at step k with a solution \(u^k\). From now on, we assume that \(F(u^k)\not =0 \) for all k and the Extragradient Algorithm generates an infinite sequence.
We recall an important property of the iterative sequence \(\{u^k\}\) generated by the Extragradient Algorithm; see, e.g., [6, 9].
Proposition 3.1
We are now in the position to establish the main result of this note. The following theorem states that the sequence \(\{u^k\}\) converges weakly to a solution of VI(K, F). This result extends the Extragradient Algorithm for solving monotone VIs [7, 9] to pseudomonotone VIs.
Theorem 3.1
Assume that F is pseudomonotone on H, sequentially weakly continuous and LLipschitz continuous on K. Assume also that Sol(K, F) is nonempty. Then, the sequence \(\{u^k\}\) generated by the Extragradient Algorithm converges weakly to a solution of VI(K, F).
Proof
Remark 3.2
It is also worth stressing that, the basic extragradient method can serve as an adequate solution method for solving pseudomonotone VIs, which was not guaranteed by the method studied in [13].
Remark 3.3
If we replace the Lipschitz continuity of F on K by its Lipschitz continuity on the whole space H, then the conclusion in Theorem 3.1 still holds for the subgradient extragradient method [7]. Indeed, a careful reviewing shows that Lemma 5.2 in [7] is also guaranteed for pseudomonotone mappings instead of monotone ones (see also [12]). The conclusion can be obtained by using a similar technique as in Theorem 3.1.
Remark 3.4
4 An Illustrative Example
In this section, we present an example to illustrate the main results obtained in Sect. 3. Another example can be found in [9, Example 5.2], where the mapping F is monotone and Lipchitz continuous. The following example is considered in [13], where the mapping F is pseudomonotone but not monotone.
5 Conclusions
We have considered the extragradient method for solving infinitedimensional variational inequalities with a pseudomonotone and Lipschitz continuous mapping. We have shown that the iterative sequence generated by the extragradient method converges weakly to a solution of the considered variational inequality, provided that such a solution exists. The strong convergence of the iteration sequence is still an open question that could be an interesting topic for a future research.
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). The author wishes to express his gratitude to the EditorinChief, the two anonymous referees and Prof. Jean Jacques Strodiot for their detailed comments and useful suggestions that allowed to improve significantly the presentation of this paper. This research is supported by the Austrian Science Foundation (FWF) under Grant No. P26640N25.
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