Some resolvent methods for general variational inclusions

Definition 2.1

Definition 2.1 Brezis, 1973

If A is a maximal monotone operator on $H\text{,}$ then, for a constant $\rho >0\text{,}$ the resolvent operator associated with A is defined by $$J_A(u)=(I+\rho A){\hspace{0.17em}}^{-1}(u)\text{,}\forall u\in H\text{,}$$ where I is the identity operator. It is well known that a monotone operator is maximal if and only if its resolvent operator is defined everywhere. In addition, the resolvent operator is a single-valued and nonexpansive, that is, $$\Vert J_A(u)-J_A(v)\Vert ⩽\Vert u-v\Vert \text{,}\forall u\text{,}v\in H\text{.}$$

Remark 2.1

It is well known that the subdifferential $\partial \phi$ of a proper, convex and lower semicontinuous function $\phi :H\to R\cup \{+\infty \}$ is a maximal monotone operator, we denote by $$J_{\phi }(u)=(I+\rho \partial \phi ){\hspace{0.17em}}^{-1}(u)\text{,}\mathrm{for}\mathrm{all}u\in H\text{,}$$ the resolvent operator associated with $\partial \phi$ , which is defined everywhere on H.

Lemma 2.1

Lemma 2.1 Brezis, 1973

For a given $z\in H$ , $u\in H$ satisfies the inequality

$$\langle u-z\text{,}v-u\rangle +\rho \phi (v)-\rho \phi (u)⩾0\text{,}\forall v\in H\text{,}$$ if and only if

(2.10)

$$u=J_{\phi }z\text{,}$$ where $J_{\phi }=(I+\rho \partial \phi ){\hspace{0.17em}}^{-1}$ is the resolvent operator and $\rho >0$ is a constant. This property of the resolvent operator $J_{\phi }$ plays an important part in our results.

Definition 2.1

The operator $T:H\to H$ is said to be

1. g-monotone, iff $$\langle \mathit{Tu}-\mathit{Tv}\text{,}g(u)-g(v)\rangle ⩾0\text{,}\forall u\text{,}v\in H\text{.}$$

2. g-pseudomonotone with respect to the function $\phi (.)\text{,}$ iff $$\begin{array}{cc}\hfill & \langle \mathit{Tu}\text{,}g(v)-g(u)+\phi (g(v))-\phi (g(u))\rangle ⩾0\hfill \\ \hfill & \mathrm{implies}\hfill \\ \hfill & \langle \mathit{Tv}\text{,}g(v)-g(u)\rangle +\phi (g(v))-\phi (g(u))⩾0\forall u\text{,}v\in H\text{.}\hfill \end{array}$$

   For $g\equiv I\text{,}$ Definition 2.1 reduces to the usual definition of monotonicity, and pseudomonotonicity of the operator T. Note that monotonicity implies pseudomonotonicity but the converse is not true, see (Alvarez and Attouch, 2001).

Lemma 3.1

Lemma 3.1 Noor, 1988

The function $u\in H$ is a solution of (2.1) if and only if $u\in H$ satisfies the relation

$$g(u)=J_A[g(u)-\rho \mathit{Tu}]\text{,}$$ where $\rho >0$ is a constant.

Algorithm 3.1

For a given $u_0\in H\text{,}$ compute the approximate solution $u_{n+1}$ by the iterative scheme $$g(u_{n+1})=J_A[g(u_n)-\rho {\mathit{Tu}}_n]\text{,}n=0\text{,}1\text{,}2\dots$$ Using the technique of Noor et al. (1993), one can prove that Algorithm 3.1 has the local convergence behaviour, which enables us to identify accurately the optimal constraint after finitely many iterations.

It is well known (Noor, 2001a,b) that the convergence of the classical resolvent methods for solving variational inclusions requires that both the operators T and g must be strongly monotone and Lipschitz continuous. These strict conditions rule out many applications of projection type methods. These factors motivated to modify the resolvent type methods by using the technique of updating the solution. Noor (2000, 2004) has used the technique of updating the solution to suggest and analyze a class of resolvent-splitting algorithms for general variational inclusions (2.1). Let $g^{-1}$ be the inverse of the operator $g\text{.}$ Then Eq. (3.1) can be written in the form: $$\begin{array}{cc}\hfill & g(y)=J_A[g(u)-\rho \mathit{Tu}]\hfill \\ \hfill & g(w)=J_A[g(y)-\rho \mathit{Ty}]\hfill \\ \hfill & g(u)=J_A[g(w)-\rho \mathit{Tw}]\text{.}\hfill \end{array}$$ or equivalently $$g(u)=J_A[I-\rho {\mathit{Tg}}^{-1}]J_A[I-\rho {\mathit{Tg}}^{-1}]J_A[I-\rho {\mathit{Tg}}^{-1}]g(u)=(I+\rho {\mathit{Tg}}^{-1}){\hspace{0.17em}}^{-1}\{J_A[I-\rho {\mathit{Tg}}^{-1}]J_A[I-\rho {\mathit{Tg}}^{-1}]J_A[I-\rho {\mathit{Tg}}^{-1}]+\rho {\mathit{Tg}}^{-1}\}g(u)\text{.}$$ Using Lemma 3.1, one can easily show that $u\in H$ is a solution of (2.1) if and only if $u\in H$ satisfies the equation $$g(u)-J_A[g(w)-\rho \mathit{Tw}]=0\text{.}$$ These modified fixed-point formulations allow us to suggest and analyze the following resolvent-splitting-type methods for solving general variational inclusions (2.1).

Algorithm 3.2

For a given $u_0\in H\text{,}$ compute the approximate solution $u_{n+1}$ by the iterative schemes $$\begin{array}{cc}\hfill & g(y_n)=J_A[g(u_n)-\rho {\mathit{Tu}}_n]\hfill \\ \hfill & g(w_n)=J_A[g(y_n)-\rho {\mathit{Ty}}_n]\hfill \\ \hfill & g(u_{n+1})=J_A[g(w_n)-\rho {\mathit{Tw}}_n]\text{,}n=0\text{,}1\text{,}2\text{,}\dots \hfill \end{array}$$ Algorithm 3.2 is known as the predictor-corrector method and is due to Moudafi and Thera (1997) and Noor (1998).

Algorithm 3.3

For a given $u_0\in H\text{,}$ compute the approximate solution $u_{n+1}$ by the iterative scheme $$g(u_{n+1})=J_A[I-\rho {\mathit{Tg}}^{-1}]J_A[I-\rho {\mathit{Tg}}^{-1}]J_A[I-\rho {\mathit{Tg}}^{-1}]g(u_n)\text{,}n=0\text{,}1\text{,}2\dots$$ which is known as the three-step forward-backward resolvent-splitting algorithm, in which the order of T and $J_A$ has not been changed unlike in Glowinski and Tallec (1989). Algorithm 3.3 is similar to that of Glowinski and Tallec (1989), which they suggested by using the Lagrange multipliers method. For the convergence analysis of Algorithms 3.3, see (Glowinski and Tallec, 1989; He and Liao, 2002; Noor, 2001a, 2004).

Algorithm 3.4

For a given $u_0\in H\text{,}$ compute $u_{n+1}$ by the iterative scheme $$g(u_{n+1})=(I+\rho {\mathit{Tg}}^{-1}){\hspace{0.17em}}^{-1}\{J_A[I-\rho {\mathit{Tg}}^{-1}]J_A[I-\rho {\mathit{Tg}}^{-1}]J_A[I-\rho {\mathit{Tg}}^{-1}]+\rho {\mathit{Tg}}^{-1}\}g(u_n)\text{,}n=0\text{,}1\text{,}2\text{,}\dots$$ which is also called the splitting resolvent method. Algorithm 3.4 can be viewed as a generalization of the splitting algorithm of Tseng (2000). Using the technique of Tseng (2000), one can discuss its applications in mathematical programming and optimization.

In this paper, we suggest an other class of self-adaptive resolvent iterative methods with the line search strategy. These methods include some known splitting methods as special cases.

We now define the resolvent residue vector by the relation $$R(u)=g(u)-J_A[g(y)-\rho \mathit{Ty}]=g(u)-J_A[J_A[g(u)-\rho \mathit{Tu}]-\rho {\mathit{Tg}}^{-1}{J}_A[g(u)-\rho \mathit{Tu}]]\text{.}$$ From Lemma 3.1, it is clear the $u\in H$ is a solution of (2.1) if and only if $u\in H$ is a zero of the equation $$R(u)=0\text{.}$$ Consider $$g(z)=(1-\eta )g(u)+\eta P_A[g(y)-\rho \mathit{Ty}]=g(u)-\eta R(u)\in H\text{.}$$ We now prove that the variational inclusion (2.1) is equivalent to the resolvent Eq. (2.11) by invoking Lemma 3.1. and this is the prime motivation of our next result.

Theorem 3.1

The variational inclusion (2.1) has a solution $u\in H$ if and only if, the resolvent Eq. (2.11) has a solution $z\in H$ , where

$$g(u)=J_{A}z$$

(3.3)

$$z=g(u)-\rho \mathit{Tu}\text{,}$$ where $J_A$ is the resolvent operator and $\rho >0$ is a constant.

Algorithm 3.5

For a given $u_0\in H\text{,}$ compute the approximate solution $u_{n+1}$ by the iterative schemes

• Predictor step

  $$\begin{array}{cc}\hfill & g(y_n)=J_A[g(u_n)-{\rho }_n{\mathit{Tu}}_n]\hfill \\ \hfill & g(w_n)=J_A[g(y_n)-{\rho }_n{\mathit{Ty}}_n]\hfill \\ \hfill & g(z_n)=g(u_n)-{\eta }_{n}R(u_n)\text{,}\hfill \end{array}$$ where ${\eta }_n=a^{m_n}$ , and $m_n$ is the smallest nonnegative integer m such that $${\eta }_n{\rho }_n\langle {\mathit{Tu}}_n-{\mathit{Tg}}^{-1}(g(u_n)-{\eta }_{n}R(u_n))\text{,}R(u_n)\rangle ⩽\sigma \Vert R(u_n){\Vert }^2\text{,}\sigma \in [0\text{,}1]\text{.}$$

• Corrector step

  $$\begin{array}{cc}\hfill & g(u_{n+1})=J_A[g(u_n)-{\alpha }_{n}d(u_n)]\text{,}n=0\text{,}1\text{,}2\text{,}\dots \hfill \\ \hfill & d(u_n)={\eta }_{n}R(u_n)+{\rho }_n{\mathit{Tz}}_n\equiv {\eta }_{n}R(u_n)+{\rho }_n{\mathit{Tg}}^{-1}(g(u_n)-{\eta }_{n}R(u_n))\hfill \\ \hfill & {\alpha }_n=\frac{{\eta }_n\langle R(u_n)\text{,}D(u_n)\rangle }{\Vert d(u_n){\Vert }^2}\hfill \\ \hfill & D(u_n)=R(u_n)-{\rho }_n{\mathit{Tu}}_n+\rho {\mathit{Tz}}_n\hfill \\ \hfill & =R(u_n)-{\rho }_n{\mathit{Tu}}_n+{\rho }_n{\mathit{Tg}}^{-1}(g(u_n)-{\eta }_{n}R(u_n))\text{,}\hfill \end{array}$$ Here ${\alpha }_n$ is the corrector step size which involves the resolvent Eq. (2.11).

  If $g\equiv I\text{,}$ the identity operator, then Algorithm 3.5 collapses to the following method for solving classical variational inclusions (2.2)

Algorithm 3.6

For a given $u_0\in H\text{,}$ compute $u_{n+1}$ by the iterative schemes

• Preditor step

  $$\begin{array}{cc}\hfill & y_n=J_A[u_n-{\rho }_n{\mathit{Tu}}_n]\hfill \\ \hfill & w_n=J_A[y_n-{\rho }_n{\mathit{Ty}}_n]\hfill \\ \hfill & z_n=u_n-{\eta }_{n}R(u_n)\text{,}\hfill \end{array}$$ where ${\eta }_n$ satisfies $${\eta }_n{\rho }_n\langle {\mathit{Tu}}_n-T(u_n-{\eta }_{n}R(u_n))\text{,}R(u_n)\rangle ⩽\sigma \Vert R(u_n){\Vert }^2\text{,}\sigma \in [0\text{,}1]$$

• Corrector step

  $$u_{n+1}=J_A[u_n-{\alpha }_n{d}_1(u_n)]\text{,}n=0\text{,}1\text{,}2\text{,}\dots$$ where $$\begin{array}{cc}\hfill & d_1(u_n)=R(u_n)+{\rho }_n{\mathit{Tz}}_n\equiv R(u_n)+{\rho }_{n}T(u_n-{\eta }_{n}R(u_n))\hfill \\ \hfill & {\alpha }_n=\frac{R(u_n)\text{,}{D}_1(u_n)\rangle }{\Vert d_1(u_n){\Vert }^2}\hfill \\ \hfill & d_1(u_n)={\eta }_{n}R(u_n)+{\rho }_{n}T(u_n-{\eta }_{n}R(u_n))\hfill \\ \hfill & D_1(u_n)=R(u_n)-{\rho }_n{\mathit{Tu}}_n+{\rho }_{n}T(u_n-{\eta }_{n}R(u_n))\text{.}\hfill \end{array}$$ Here ${\alpha }_n$ is the corrector step size, which involves the resolvent equation.

  For ${\eta }_n=1$ , Algorithm 3.5 becomes:

Algorithm 3.7

For a given $u_0\in H\text{,}$ compute $u_{n+1}$ by the iterative schemes

• Predictor step

  $$\begin{array}{cc}\hfill & g(y_n)=J_A[g(u_n)-{\rho }_n{\mathit{Tu}}_n]\hfill \\ \hfill & g(w_n)=J_A[g(y_n)-{\rho }_n{\mathit{Ty}}_n]\text{,}\hfill \end{array}$$ where ${\rho }_n$ satisfies $${\rho }_n\langle {\mathit{Tu}}_n-{\rho }_n{\mathit{Tw}}_n\text{,}R(u_n)\rangle ⩽\sigma \Vert R(u_n){\Vert }^2\text{,}\sigma \in (0\text{,}1)\text{.}$$

• Corrector step

  $$g(u_{n+1})=J_A[g(u_n)-{\alpha }_n{d}_2(u_n)]\text{,}n=0\text{,}1\text{,}2\text{,}\dots \text{,}$$ where $$\begin{array}{cc}\hfill & d_2(u_n)=R(u_n)+{\rho }_n{\mathit{Tw}}_n\hfill \\ \hfill & {\alpha }_n=\frac{\langle R(u_n)\text{,}{D}_2(u_n)\rangle }{\Vert d_2(u_n){\Vert }^2}\hfill \\ \hfill & D_2(u_n)=R(u_n)-{\rho }_n{\mathit{Tu}}_n-{\rho }_n{\mathit{Tw}}_n\hfill \end{array}$$ and ${\alpha }_n$ is the corrector step size.

  If $A(.)\equiv \partial \phi (.)$ , where $\partial \phi$ is the subdifferential of a proper, convex and lower semicontinuous function $\phi :H\to R\cup \{+\infty \}$ , then $J_A\equiv J_{\phi }=(I+\partial \phi ){\hspace{0.17em}}^{-1}$ , the resolvent operator and consequently Algorithm 3.5 collapses to:

Algorithm 3.8

For a given $u_0\in H\text{,}$ compute the approximate solution $u_{n+1}$ by the iterative schemes

• Predictor step

  (3.7)

  $$g(y_n)=J_{\phi }[g(u_n)-{\rho }_n{\mathit{Tu}}_n]$$

  (3.8)

  $$g(w_n)=J_{\phi }[g(y_n)-{\rho }_n{\mathit{Ty}}_n]$$

  (3.9)

  $$g(z_n)=g(u_n)-{\eta }_{n}R(u_n)\text{,}$$ where ${\eta }_n=a^{m_n}\text{,}$ and $m_n$ is the smallest nonnegative integer m such that

  (3.10)

  $${\eta }_n{\rho }_n\langle {\mathit{Tu}}_n-{\mathit{Tg}}^{-1}(g(u_n)-{\eta }_{n}R(u_n))\text{,}R(u_n)\rangle ⩽\sigma \Vert R(u_n){\Vert }^2\text{,}\sigma \in [0\text{,}1]\text{.}$$

• Corrector step

  (3.11)

  $$g(u_{n+1})=J_{\phi }[g(u_n)-{\alpha }_{n}d(u_n)]\text{,}n=0\text{,}1\text{,}2\text{,}\dots$$

  (3.12)

  $$\begin{array}{cc}\hfill & d_3(u_n)={\eta }_{n}R(u_n)+{\rho }_n{\mathit{Tz}}_n\equiv {\eta }_{n}R(u_n)+{\rho }_n{\mathit{Tg}}^{-1}(g(u_n)\hfill \\ \hfill & -{\eta }_{n}R(u_n))\hfill \end{array}$$

  (3.13)

  $${\alpha }_n=\frac{{\eta }_n\langle R(u_n)\text{,}{D}_3(u_n)\rangle }{\Vert d_3(u_n){\Vert }^2}$$

  (3.14)

  $$D_3(u_n)=R(u_n)-{\rho }_n{\mathit{Tu}}_n+{\rho }_n{\mathit{Tg}}^{-1}(g(u_n)-{\eta }_{n}R(u_n))\text{,}$$ Here ${\alpha }_n$ is the corrector step size which involves the resolvent equation.

  For ${\alpha }_n=1\text{,}$ Algorithm 3.5 is exactly the resolvent-splitting Algorithm 3.1, which is due to Moudafi and Thera (1997). For $g\equiv I\text{,}$ and ${\eta }_n=1\text{,}$ one can obtain several new splitting type algorithms for solving classical (quasi) variational inclusions (2.2) and (2.3). This clearly shows that Algorithm 3.5 is unifying ones and includes several new and previously known methods as special cases.

  For the convergence analysis of Algorithm 3.8, we need the following result, which can be proved using the technique of Noor (2001a). For the sake of completeness, we sketch the main ideas.

Theorem 3.1

If $\overline{u}\in H\text{,}$ is a solution of (2.1) and the operator $T:H\to H$ is g-monotone, then

$$\langle g(u)-g(\overline{u})\text{,}d(u)\rangle ⩾(\eta -\sigma )\Vert R(u){\Vert }^2\text{,}\forall u\in H\text{.}$$

Theorem 3.2

Let $\overline{u}\in H$ be a solution of (2.1) and let $u_{n+1}$ be the approximate solution obtained from Algorithm 3.8. Then

$$\Vert g(u_{n+1})-g(\overline{u}){\Vert }^2⩽\Vert g(u_n)-g(\overline{u}){\Vert }^2-\frac{({\eta }_n-\sigma ){\hspace{0.17em}}^2\Vert R(u_n){\Vert }^4}{\Vert d(u_n){\Vert }^2}\text{.}$$

Proof

From (3.15), we have $$\Vert g(u_{n+1})-g(\overline{u}){\Vert }^2⩽\Vert g(u_n)-g(\overline{u})-{\alpha }_n{d}_3(u_n){\Vert }^2⩽\Vert g(u_n)-g(\overline{u}){\Vert }^2-2{\alpha }_n\langle g(u_n)-g(\overline{u})\text{,}{d}_3(u_n)\rangle +{\alpha }_n\Vert d_3(u_n){\Vert }^2⩽\Vert g(u_n)-g(\overline{u}){\Vert }^2-{\alpha }_n\langle R(u_n)\text{,}{D}_3(u_n)⩽\Vert g(u_n)-g(\overline{u}){\Vert }^2-{\alpha }_n({\eta }_n-\sigma )\Vert R(u_n){\Vert }^2⩽\Vert g(u_n)-g(\overline{u}){\Vert }^2-\frac{({\eta }_n-\sigma ){\hspace{0.17em}}^2\Vert R(u_n){\Vert }^4}{\Vert d_3(u_n){\Vert }^2}\text{.}$$ the required results.  □

Theorem 3.3

Let $\overline{u}\in H$ be a solution of (2.1) and let $u_{n+1}$ be the approximate solution obtained from Algorithm 3.8. If H is a finite dimensional space and g is injective, then ${\lim }_{n\to \infty }(u_n)=\overline{u}\text{.}$

Proof

Let $\overline{u}$ be a solution of (2.1). Then, from (3.16), it follows that the sequence $\{\Vert g(u_n)-g(\overline{u})\Vert \}$ is nonincreasing and consequently the sequence $\{g(u_n)\}$ is bounded. Thus, under the assumptions of $g$ , it follows that the sequence $\{u_n\}$ is bounded. Furthermore, from (3.16), we have $$\sum _{n=0}^{\infty }\frac{({\eta }_n-\sigma ){\hspace{0.17em}}^2\Vert R(u_n){\Vert }^4}{\Vert d(u_n){\Vert }^2}⩽\Vert g(u_0)-g(\overline{u}){\Vert }^2\text{,}$$ which implies that

$$\underset{n\to \infty }{\lim }R(u_n)=0\text{,}$$ or

(3.18)

$$\underset{n\to \infty }{\lim }{\eta }_n=0\text{.}$$ Assume that (3.17) holds. Let $\overline{u}$ be the cluster point of $\{u_n\}$ and let the subsequence $\{u_{n_j}\}$ of the sequence of $\{u_n\}$ converge to $\overline{u}\text{.}$ Since T and g are continuous, it follows that R is continuous and $$R(\overline{u})=\underset{n_j\to \infty }{\lim }R(u_{n_j})=0\text{,}$$ from which it follows that $\overline{u}$ is a solution of (2.1) by invoking Lemma 3.1 and

(3.19)

$$\Vert g(u_{n+1})-g(\overline{u}){\Vert }^2⩽\Vert g(u_n)-g(\overline{u}){\Vert }^2\text{.}$$ Thus the sequence $\{u_n\}$ has exactly one cluster point and consequently $$\underset{n\to \infty }{\lim }g(u_n)=g(\overline{u})\text{.}$$ Since g is injective, it follows that ${\lim }_{n\to \infty }{u}_n=\overline{u}\in H$ satisfying the general variational inclusion (2.1).

Assume that (3.18) holds, that is, ${\lim }_{n\to \infty }{\eta }_n=0\text{.}$ If (3.10) does not hold, then by a choice of ${\eta }_n$ , we obtain

$$\sigma \Vert R(u_n){\Vert }^2⩽{\rho }_n{\eta }_n\langle {\mathit{Tu}}_n-{\mathit{Tg}}^{-1}(g(u_n)-{\eta }_{n}R(u_n))\text{,}R(u_n)\rangle \text{.}$$ Let $\overline{u}$ be a cluster point of $\{u_n\}$ and let $\{u_{n_j}\}$ be the corresponding subsequence of $\{u_n\}$ converging to $\overline{u}\text{.}$ Taking the limit in (3.20), we have $\sigma \Vert R(\overline{u}){\Vert }^2⩽$ , which implies that $R(\overline{u})=0\text{,}$ that is, $\overline{u}\in H$ is a solution of (2.1) by invoking Lemma 3.1 and (3.19) holds. Repeating the above arguments, we conclude that ${\lim }_{n\to \infty }{u}_n=\overline{u}\text{.}$  □

Algorithm 4.1

For a given $u_0\in H\text{,}$ compute the approximate solution $u_{n+1}$ by the iterative scheme $$g(u_{n+1})=J_A[g(u_n)-\rho {\mathit{Tu}}_{n+1}]\text{,}n=0\text{,}1\text{,}2\dots \text{,}$$ which is known as the implicit resolvent method. For the convergence analysis of Algorithm 4.1, see (Noor, 2001a).

We can rewrite (3.1) in the following equivalent form.

$$g(u)=J_A[g(u)-\rho \mathit{Tu}+\alpha (g(u)-g(u))]\text{,}$$ where $\rho >0$ and $\alpha >0$ are constants.

This formulation is used to suggest the following iterative method for solving the general variational inclusions (2.1).

Algorithm 4.2

For a given $u_0\in H\text{,}$ compute the approximate solution $u_{n+1}$ by the iterative scheme $$g(u_{n+1})=J_A[g(u_n)-\rho {\mathit{Tu}}_{n+1}+{\alpha }_n(g(u_n)-g(u_{n-1}))]\text{,}n=0\text{,}1\text{,}2\text{,}\dots$$ If $A(.)\equiv \partial \phi (.)$ , where $\partial \phi$ is the subdifferential of a proper, convex and lower semicontinuous function $\phi :H\to R\cup \{+\infty \}$ , then $J_A\equiv J_{\phi }=(I+\partial \phi ){\hspace{0.17em}}^{-1}$ , the resolvent operator and consequently Algorithm 4.2 collapses to:

Algorithm 4.3

For a given $u_0\in H\text{,}$ compute the approximate solution $u_{n+1}$ by the iterative scheme $$g(u_{n+1})=J_{\phi }[g(u_n)-\rho {\mathit{Tu}}_{n+1}+{\alpha }_n(g(u_n)-g(u_{n-1}))]\text{,}n=0\text{,}1\text{,}2\text{,}\dots$$ Using Lemma 3.1, Algorithm 4.3 can be written as

Algorithm 4.4

For a given $u_0\in H\text{,}$ compute the approximate solution $u_{n+1}$ by the iterative scheme

$$\langle \rho {\mathit{Tu}}_{n+1}+g(u_{n+1})-g(u_n)-{\alpha }_n(g(u_n)-g(u_{n-1}))\text{,}g(v)-g(u_{n+1})\rangle +\rho \phi (g(v))-\rho \phi (g(u_{n+1}))⩾0\text{,}\forall v\in H\text{.}$$ If $g=I\text{,}$ the identity operator, then Algorithm 4.3 reduces to:

Algorithm 4.5

For a given $u_0\in H\text{,}$ compute the approximate solution $u_{n+1}$ by the iterative scheme $$\langle \rho {\mathit{Tu}}_{n+1}+u_{n+1}-u_n-{\alpha }_n(u_n-u_{n-1})\text{,}v-u_{n+1}\rangle +\rho \phi (v)-\rho \phi (u_{n+1})⩾0\text{,}\forall v\in H\text{.}$$ In a similar way, one can suggest and analyze several iterative methods for solving the general variational inclusions and its related optimization.

For the convergence analysis of Algorithm 4.4, we recall the following well known result.

Lemma 4.1

$$2\langle u\text{,}v\rangle =\Vert u+v{\Vert }^2-\Vert u{\Vert }^2-\Vert v{\Vert }^2\text{,}\forall u\text{,}v\in H\text{.}$$

Theorem 4.1

Let $\overline{u}\in H$ be a solution of (2.1) and let $u_{n+1}$ be the approximate solution obtained from Algorithm 4.4 If the operator $T:H\to H$ is g-pseudomonotone, then

$$\Vert g(u_{n+1})-g(\overline{u}){\Vert }^2⩽\Vert g(u_n)-g(\overline{u}){\Vert }^2+{\alpha }_n\{\Vert g(u_n)-g(\overline{u}){\Vert }^2-\Vert g(\overline{u})-g(u_{n-1}){\Vert }^2+2\Vert g(u_n)-g(u_{n-1}){\Vert }^2\}-\Vert g(u_{n+1})-g(u_n)-{\alpha }_n(g(u_n)-g(u_{n-1})){\Vert }^2\text{.}$$

Proof

Let $\overline{u}\in H$ be a solution of (2.1). Then $$\langle T\overline{u}\text{,}g(v)-g(\overline{u})\rangle +\phi (g(v))-\phi (g(\overline{u}))⩾0\text{,}\forall v\in H\text{,}$$ implies that

$$\langle \mathit{Tv}\text{,}g(v)-g(\overline{u})+\phi (g(v))-\phi (g(\overline{u}))\rangle ⩾0\text{,}\forall v\in H\text{,}$$ since T is g-pseudomonotone.

Taking $v=u_{n+1}$ in (4.4), we have

$$\langle {\mathit{Tu}}_{n+1}\text{,}g(u_{n+1})-g(\overline{u})\rangle +\phi (g(u_{n+1}))-\phi (g(\overline{u}))⩾0\text{.}$$ Setting $v=\overline{u}$ in (4.2), we obtain

(4.6)

$$\langle \rho {\mathit{Tu}}_{n+1}+g(u_{n+1})-g(u_n)-{\alpha }_n\{g(u_n)-g(u_{n-1})\}\text{,}g(\overline{u})-g(u_{n+1})\rangle +\rho \phi (g(\overline{u}))-\rho \phi (g(u_{n+1}))⩾0\text{.}$$ Combining Eqs. (4.5) and (4.6), we have

(4.7)

$$\langle g(u_{n+1})-g(u_n)-{\alpha }_n\{g(u_n)-g(u_{n-1})\}\text{,}g(\overline{u})-g(u_{n+1})\rangle ⩾0\text{,}$$ which can be written as

(4.8)

$$\langle g(u_{n+1})-g(u_n)\text{,}g(\overline{u})-g(u_{n+1})\rangle ⩾{\alpha }_n\langle g(u_n)-g(u_{n-1})\text{,}g(\overline{u})-g(u_n)+g(u_n)-g(u_{n+1})\rangle \text{.}$$ Using Lemma 4.1 and rearranging the terms of (4.8), one can easily obtain (4.3), the required result.  □