Convergence analysis of projection method for variational inequalities

Abstract

The main contributions of this paper are the proposition and the convergence analysis of a class of inertial projection-type algorithm for solving variational inequality problems in real Hilbert spaces where the underline operator is monotone and uniformly continuous. We carry out a unified analysis of the proposed method under very mild assumptions. In particular, weak convergence of the generated sequence is established and nonasymptotic O(1 / n) rate of convergence is established, where n denotes the iteration counter. We also present some experimental results to illustrate the profits gained by introducing the inertial extrapolation steps.

Appendix

In this case, we present a version of Lemma 4.3 for the case when F is pseudo-monotone.

Lemma 8.1

Let F be pseudo-monotone, uniformly continuous and sequentially weakly continuous on H. Assume that Assumption 3.2 holds. Furthermore, let $\{x_{n_k}\}$ be a subsequence of $\{x_n\}$ converging weakly to a limit point p. Then, $p \in \text {SOL}$.

Proof

By the definition of $z_{n_k}$ together with (6), we have

$$\begin{aligned} \langle w_{n_k}-F(w_{n_k})-z_{n_k},x-z_{n_k}\rangle \le 0, \quad \forall \, x\in C, \end{aligned}$$

which implies that

$$\begin{aligned} \langle w_{n_k}-z_{n_k},x-z_{n_k}\rangle \le \langle F(w_{n_k}),x-z_{n_k} \rangle ,\quad \forall \, x \in C. \end{aligned}$$

Hence,

$$\begin{aligned} \langle w_{n_k}-z_{n_k},x-z_{n_k}\rangle +\langle F(w_{n_k}),z_{n_k}-w_{n_k}\rangle \le \langle F(w_{n_k}),x-w_{n_k}\rangle ,\quad \forall \, x\in C.\qquad \end{aligned}$$

(39)

Fix $x \in C$ and let $k\rightarrow \infty$ in (39). Since $\lim _{k \rightarrow \infty } \Vert w_{n_k}-z_{n_k}\Vert =0$, we have

$$\begin{aligned} 0\le \liminf _{k \rightarrow \infty } \langle F(w_{n_k}),x-w_{n_k}\rangle \end{aligned}$$

(40)

for all $x \in C$. Now, we choose a sequence $\{\epsilon _k\}_k$ of positive numbers decreasing and tending to 0. For each $\epsilon _k$, we denote by $N_k$ the smallest positive integer such that

$$\begin{aligned} \left\langle F(w_{n_j}), x- w_{n_j} \right\rangle + \epsilon _k \ge 0 \quad \forall j \ge N_k, \end{aligned}$$

(41)

where the existence of $N_k$ follows from (40). Since $\left\{ \epsilon _k \right\}$ is decreasing, it is easy to see that the sequence $\left\{ N_k \right\}$ is increasing. Furthermore, for each k, $F(w_{N_k}) \not = 0$ and, setting

$$\begin{aligned} v_{N_k}=\frac{F(w_{N_k})}{\Vert F(w_{N_k}\Vert ^2}, \end{aligned}$$

we have $\left\langle F(w_{N_k}),v_{N_k} \right\rangle =1$ for each k. Now we can deduce from (41) that for each k

$$\begin{aligned} \left\langle F(w_{N_k}),x+\epsilon _k v_{N_k} -w_{N_k} \right\rangle \ge 0, \end{aligned}$$

and, since F is pseudo-monotone, that

$$\begin{aligned} \left\langle F(x+\epsilon _k v_{N_k}),x+\epsilon _k v_{N_k} -w_{N_k} \right\rangle \ge 0. \end{aligned}$$

(42)

On the other hand, we have that $\left\{ x_{n_k} \right\}$ converges weakly to p when $k \rightarrow \infty$. Since F is sequentially weakly continuous on C, $\left\{ F(w_{n_k}) \right\}$ converges weakly to F(p). We can suppose that $F(p) \not = 0$ (otherwise, p is a solution). Since the norm mapping is sequentially weakly lower semicontinuous, we have

$$\begin{aligned} 0<\Vert F(p) \Vert \le \lim \inf _{k \rightarrow \infty } \Vert F(w_{n_k})\Vert . \end{aligned}$$

Since $\left\{ w_{N_k} \right\} \subset \left\{ w_{n_k} \right\}$ and $\epsilon _k \rightarrow 0$ as $k \rightarrow \infty$, we obtain

$$\begin{aligned} 0\le & {} \lim \sup _{k \rightarrow \infty } \Vert \epsilon _k v_{N_k}\Vert = \lim \sup _{k \rightarrow \infty } \Big (\frac{\epsilon _k}{\Vert F(w_{n_k})\Vert }\Big )\\\le & {} \frac{\lim \sup _{k \rightarrow \infty }\epsilon _k}{\lim \inf _{k \rightarrow \infty }\Vert F(w_{n_k})\Vert } \le \frac{0}{\Vert F(p)\Vert }=0, \end{aligned}$$

which implies that $\lim _{k \rightarrow \infty } \Vert \epsilon _k v_{N_k}\Vert =0$. Hence, taking the limit as $k \rightarrow \infty$ in (42), we obtain

$$\begin{aligned} \left\langle F(x),x-p \right\rangle \ge 0. \end{aligned}$$

Now, using Lemma 2.2 of Mashreghi and Nasri (2010), we have that $p\in \text {SOL}$. $\square$