 Research
 Open Access
 Published:
Generalized proximaltype methods for weak vector variational inequality problems in Banach spaces
Fixed Point Theory and Applications volume 2015, Article number: 191 (2015)
Abstract
In this paper, we propose a class of generalized proximaltype method by the virtue of Bregman functions to solve weak vector variational inequality problems in Banach spaces. We carry out a convergence analysis on the method and prove the weak convergence of the generated sequence to a solution of the weak vector variational inequality problems under some mild conditions. Our results extend some known results to more general cases.
Introduction
The wellknown proximal point algorithm (PPA, for short) is a powerful tool for solving optimization problems and variational inequality problems, which was first introduced by Martinet [1] and its celebrated progress was attained in the work of Rockafellar [2]. The classical proximal point algorithm generated a sequence \(\{z^{k}\}\subset H\) with an initial point \(z^{0}\) through the following iteration:
where \(\{c_{k}\}\) is a sequence of positive real numbers bounded away from zero. Rockafellar [2] proved that for a maximal monotone operator T, the sequence \(\{z^{k}\}\) weakly converges to a zero of T under some mild conditions. From then on, many works have been devoted to the investigation of the proximal point algorithms, its applications, and generalizations (see [3–8] and the references therein for scalarvalued optimization problems and variational inequality problems; see [9–14] for vectorvalued optimization problems). The notion of a Bregman function has its origin in [15] and the name was first used in optimization problems and its related topics by Censor and Lent in [16]. Bregman function algorithms have been extensively used for various optimization problems and variational inequality problems (e.g. [17–20] in finitedimensional spaces, [21, 22] in infinitedimensional spaces).
The concept of vector variational inequality was firstly introduced by Giannessi [23] in finitedimensional spaces. The vector variational inequality problems have found a lot of important applications in multiobjective decision making problems, network equilibrium problems, traffic equilibrium problems, and so on. Because of these significant applications, the study of vector variational inequalities has attracted wide attention. Chen and Yang [24] investigated general vector variational inequality problems and vector complementary problems in infinitedimensional spaces. Chen [25] considered vector variational inequality problems with a variable ordering structure. Through the last 20 years of development, existence results of solutions, duality theorems, and topological properties of solution sets of several kinds of vector variational inequalities have been derived. One can find a fairly complete review of the main results as regards the vector variational inequalities in the monograph [26]. Very recently, Chen [27] constructed a class of vectorvalued proximaltype method for solving a weak vector variational inequality problem in finitedimensional spaces. They generalized some the classical results of Rockafellar [2] from the scalarvalued case to the vectorvalued case.
An important motivation for making an analysis of the convergence properties of both proximal point algorithms and Bregman function algorithms is related to the mesh independence principle [28–30]. The mesh independence principle relies on infinite dimensional convergence results for predicting the convergence properties of the discrete finitedimensional method. Furthermore it can provide a theoretical foundation for the justification of refinement strategies and help to design the refinement process. As we know, many practical problems in economics and engineering are modeled in infinitedimensional spaces, thus it is necessary to analyze the convergence property of the algorithms in abstract spaces. Motivated by [22, 27], in this paper we propose a class of generalized proximaltype methods by virtue of the Bregman function for solving weak vector variational inequality problems in Banach spaces. We carry out a convergence analysis of the method and prove convergence of the generated sequence to a solution of the weak vector variational inequality problems under some mild conditions.
The paper is organized as follows.
In Section 2, we present some basic concepts, assumptions and preliminary results. In Section 3, we introduce the proximaltype method and carry out convergence analysis on the method. In Section 4, we draw a conclusion and make some remarks.
Preliminaries
In this section, we present some basic definitions and propositions for the proof of our main results.
Let X be a smooth reflexive Banach space with norm \(\Vert \cdot \Vert _{X}\), denote by \(X^{*}\) the dual space of X and Y is also a smooth reflexive Banach space ordered by a nontrivial, closed and convex cone C with nonempty interior intC, which defines a partial order \(\leq_{C}\) in Y, i.e., \(y_{1}\leq_{C} y_{2}\) if and only if \(y_{2}y_{1}\in C\) for any \(y_{1}, y_{2}\in Y\) and for relation \(\leq_{\operatorname {int}C}\), given by \(y_{1}\leq_{\operatorname {int}C} y_{2}\) if and only if \(y_{2}y_{1}\in \operatorname {int}C\) for any \(y_{1}, y_{2}\in Y\). We denote by \(Y^{*}\) the dual space of Y, by \(C^{*}\) the positive polar cone of C, i.e.,
We denote \(C^{*+}=\{z\in Y^{*}: \langle y, z\rangle>0\}\) for all \(y\in C\backslash\{0\}\).
Lemma 2.1
[31]
Let \(e\in \operatorname {int}C\) be fixed and \(C^{0}=\{z\in C^{*}\mid \langle z, e\rangle=1\}\), then \(C^{0}\) is a \(weak^{*}\)compact subset of \(C^{*}\).
Denote by \(L(X, Y)\) the set of all continuous linear operators from X to Y. Denote by \(\langle \varphi , x\rangle\) the value \(\varphi (x)\) of φ at \(x\in X\) if \(\varphi \in L(X,Y)\). Let \(X_{0}\subset X\) be nonempty, closed, and convex, consider the weak vector variational inequality problem of finding \(\bar {x}\in X_{0}\) such that
where \(T: X_{0}\rightarrow L(X, Y)\) be a mapping. Denote by X̄ the solution set of (VVI). Let \(\lambda\in C^{0}\) and \(\lambda(T): X_{0}\rightarrow X^{*}\), consider the corresponding scalarvalued variational inequality problem of finding \(x^{*}\in X_{0}\) such that
Denote by \(\bar{X}_{\lambda}\) the solution set of (VIP_{ λ }).
Definition 2.1
[32]
Let \(X_{0}\subset X\) be nonempty, closed, and convex, and \(F: X_{0}\rightarrow X^{*}\) be a singlevalued mapping.

(i)
F is said to be monotone on \(X_{0}\) if, for any \(x_{1}, x_{2}\in X_{0}\), we have
$$\bigl\langle F(x_{1})F(x_{2}), x_{1}x_{2} \bigr\rangle \geq0. $$ 
(ii)
F is said to be pseudomonotone on \(X_{0}\) if, for any \(x_{1}, x_{2}\in X_{0}\), we have
$$\bigl\langle F(x_{2}), x_{1}x_{2}\bigr\rangle \geq0\quad \Rightarrow\quad \bigl\langle F(x_{1}), x_{1}x_{2} \bigr\rangle \geq0. $$
Definition 2.2
[26]
Let \(X_{0}\subset X\) be nonempty, closed, and convex. \(T: X_{0}\rightarrow L(X, Y)\) is a mapping, which is said to be Cmonotone on \(X_{0}\) if, for any \(x_{1}, x_{2}\in X_{0}\), we have
Proposition 2.1
[26]
Let \(X_{0}\) and T be defined as in Definition 2.2, we have the following statements:

(i)
T is Cmonotone if and only if, for any \(\lambda\in C^{0}\), the mapping \(\lambda(T): X_{0}\rightarrow X^{*}\) is monotone;

(ii)
if T is Cmonotone, then for any \(\lambda\in C^{0}\), \(\lambda(T): X_{0}\rightarrow X^{*}\) is pseudomonotone.
Definition 2.3
[33]
Let \(L\subset L(X, Y)\) be a nonempty set. The weak and strong Cpolar cones of L are defined, respectively, by
and
Definition 2.4
[26]
Let \(K\subset X\) be nonempty, closed, and convex, let \(F: K\rightarrow Y\) be a vectorvalued mapping. A linear operator V is said to be a strong subgradient of F at \(\bar{x}\in K\) if
A linear operator V is said to be a weak subgradient of F at \(\bar{x}\in K\) if
Denote by \(\partial^{w}_{C}F(\bar{x})\) the set of weak subgradients of F on K at x̄.
Let \(K\subset X\) be nonempty, closed, and convex. A vectorvalued indicator function \(\delta(x\mid K)\) of K at x is defined by
An important and special case in the theory of weak subgradients is the case that \(F(x)=\delta(x\mid K)\) becomes a vectorvalued indicator function of K. By Definition 2.4, we obtain \(V\in \partial^{w}_{C}\delta(x^{*}\mid K)\) if and only if
Definition 2.5
A set \(VN^{w}_{K}(x^{*})\subset L(X, Y)\) is said to be a weak normality operator set to K at \(x^{*}\), if for every \(V\in VN^{w}_{K}(x^{*})\), the inequality (2.3) holds.
Clearly, \(VN^{w}_{K}(x^{*})=\partial^{w}_{C}\delta(x^{*}\mid K)\). As for the scalarvalued case, from [34] we know that \(v^{*}\in\partial\delta_{K}(x^{*})=N_{K}(x^{*})\) if and only if
where \(\delta_{K}(x)\) is the scalarvalued indicator function of K. The inequality (2.4) means that \(v^{*}\) is normal to K at \({x^{*}}\).
Definition 2.6
Let \(VN^{w}_{K}(\cdot): X\rightrightarrows L(X, Y)\) be a setvalued mapping, which is said to be a weak normal mapping for K, if for any \(y\in K\), \(V\in VN^{w}_{K}(y)\) such that
\(VN^{s}_{K}(\cdot)\) is said to be a strong normal mapping for K, if for any \(y\in K\), \(V\in VN^{s}_{K}(y)\) such that
As in [35], the normal mapping for K is a setvalued mapping, which is defined, for any \(y\in K\), \(v\in N_{K}(y)\), by
Next we will introduce the definition and some basic results as regards the maximal monotone mapping.
Definition 2.7
[32]
Let a setvalued map \(G: X_{0}\subset X\rightrightarrows X^{*}\) be given, it is said to be monotone if
for all z and z̄ in \(X_{0}\), all w in \(G(z)\) and w̄ in \(G(\bar{z})\). It is said to be maximal monotone if, in addition, the graph
is not properly contained in the graph of any other monotone operator from X to \(X^{*}\).
Definition 2.8
[36]
An operator \(A : X\rightarrow X^{*}\) is said to be regular if for any \(u\in R(A)\) and for any \(y\in \operatorname {dom}(A)\),
Proposition 2.2
[37]
The subdifferential of a proper lower semicontinuous convex function ϕ is a regular operator.
Lemma 2.2
[35]
Let K be a nonempty, closed, and convex subset of X.

(a)
If \(T_{1} : X \rightrightarrows X^{*}\) is the normal mapping to K and \(T_{2}: X\rightarrow X^{*}\) is a singlevalued monotone operator such that \(\operatorname {int}K\cap \operatorname {dom}(T_{2})\neq \emptyset\) and \(T_{2}\) is continuous on K, then \(T_{1}+T_{2}\) is a maximal monotone operator.

(b)
If \(T_{1} , T_{2} : X \rightrightarrows X^{*}\) are maximal monotone operators, and \(\operatorname {int}(\operatorname {dom}T_{1}) \cap \operatorname {dom}T_{2}\neq\emptyset\), then \(T_{1}+T_{2}\) is a maximal monotone operator.
Lemma 2.3
[22]
Assume that X is a reflexive Banach space and J its normalized duality mapping. Let \(T_{0}, T_{1}: X\rightrightarrows X^{*}\) be maximal monotone operators such that

(a)
\(T_{1}\) is regular.

(b)
\(\operatorname {dom}T_{1}\cap \operatorname {dom}T_{0}\neq\emptyset\) and \(R(T_{1})=X^{*}\).

(c)
\(T_{1}+T_{0}\) is maximal monotone.
Then we have \(R(T_{1}+T_{0})=X^{*}\).
Proposition 2.3
[38]
Any maximal monotone operator \(S : X \rightrightarrows X^{*}\) is demiclosed, if the conditions below hold:

If \(\{x_{k}\}\) converges weakly to x and \(\{w_{k} \in S(x_{k})\}\) converges strongly to w, then \(w \in S(x)\).

If \(\{x_{k}\}\) converges strongly to x and \(\{w_{k} \in S(x_{k})\}\) converges weakly to w, then \(w \in S(x)\).
Definition 2.9
[22]
Let X be a smooth reflexive Banach space and \(f : X\rightarrow R \) a strictly convex, proper, and lower semicontinuous function with closed domain \(D := \operatorname {dom}(f)\). Assume from now on that \(\operatorname {int}D\neq\emptyset\) and that f is Gâteaux differentiable on intD. The Bregman distance with respect to f is the function \(B_{f}(z, x): D\times \operatorname {int}D\rightarrow R\) defined by
where \(\nabla f(x)\) is the differential of f defined in intD.
Consider the following set of assumptions on f.
 A_{1}::

The right level sets of \(B_{f} (y,\cdot )\),
$$ S_{y, \alpha}= \bigl\{ z\in \operatorname {int}D: B_{f} (y, z)\leq \alpha \bigr\} , $$(2.9)are bounded for all \(\alpha\geq0\) and for all \(y\in D\).
 A_{2}::

If \(\{x^{k}\}\subset \operatorname {int}K\) and \(\{y^{k}\}\subset \operatorname {int}K\) are bounded sequences such that \(\lim_{k\rightarrow+\infty} \Vert x_{k}y_{k}\Vert \)=0, then
$$ \lim_{k\rightarrow+\infty} \bigl(\nabla f(x_{k}) \nabla f(y_{k}) \bigr)=0. $$(2.10)  A_{3}::

Assume \(\{x^{k}\}\subset K\) is bounded, \(\{ y^{k}\}\subset \operatorname {int}K\) is such that \(w\mbox{} \lim_{k\rightarrow +\infty}y^{k}= y\). Further assume that \(w\mbox{} \lim_{k\rightarrow +\infty} B_{f}(x^{k}, y^{k})=0\), then
$$w\mbox{} \lim_{k\rightarrow+\infty}x^{k}= y. $$  A_{4}::

For every \(y\in X^{*}\), there exists \(x\in \operatorname {int}D\) such that \(\nabla f(x) = y\).
We say that f is said to be a Bregman function if it satisfies A_{1}A_{3}, and f is said to be a coercive Bregman function if it satisfies A_{1}A_{4}. Without loss of generality, we assume \(X_{0}\subset \operatorname {dom}f\).
Definition 2.10
[39]
Let \(X_{0}\subset X\) be a closed and convex set with \(\operatorname {int}X_{0}\neq\emptyset\), \(f: X_{0}\rightarrow R\) a convex function which is Gâteaux differentiable in \(\operatorname {int}X_{0}\). We define the modulus of convexity of f, \(v_{f}: \operatorname {int}X_{0}\times R_{+}\rightarrow R_{+}\) by
where \(B_{f}(x,z)\) is the Bregman distance with respect to f.

(a)
The function f is said to be totally convex in \(\operatorname {int}X_{0}\) if and only if \(v_{f}(z, t)>0\) for all \(z\in \operatorname {int}X_{0}\) and \(t>0\).

(b)
The function f is said to be uniformly totally convex if for any bounded set \(K\subset \operatorname {int}X_{0}\) and \(t>0\), we have
$$ \inf_{x\in K}v_{f}(x,t)>0. $$(2.12)
Next we will introduce some fundamental definitions of the asymptotic analysis in infinite spaces.
Definition 2.11
[40]
Let K be a nonempty set in X. Then the asymptotic cone of the set K, denoted by \(K^{\infty}\), is the set of all vectors \(d\in X\) that are weak limits in the direction of the sequence \(\{x_{k}\}\subset K\), namely
In the case that K is convex and closed, then, for any \(x_{0}\in K\),
Main results
Proposition 3.1
Let \(X_{0}\subset X\) be nonempty, closed, and convex, and \(VN^{w}_{X_{0}}(\cdot)\) be a weak normal mapping for \(X_{0}\). For any \(x^{*}\in X_{0}\), and \(\varphi\in VN^{w}_{X_{0}}(x^{*})\), there exists a \(\lambda\in C^{0}\) such that \(\lambda (\varphi)\in N_{X_{0}}(x^{*})\).
Proof
By the definition of the weak normal mapping, we know that
It follows that
and
That is,
By virtue of the convexity of \(X_{0}\) and the HahnBanach theorem, there exists a \(\lambda^{*}\in C^{*}\backslash\{0\}\) such that
Since \(\Vert \lambda^{*}\Vert >0\), one obtains
Clearly, we have \(\frac{\lambda^{*}}{\Vert \lambda^{*}\Vert }\in C^{0}\). Without loss of generality, let \(\lambda=\frac{\lambda^{*}}{\Vert \lambda^{*}\Vert }\), one has
That is, \(\lambda(\varphi)\in N_{X_{0}}(x^{*})\). The proof is complete. □
We propose the following exact generalized proximaltype method (GPTM, for short) for solving the problem (WVVI):
Step (1) take \(x_{0}\in X_{0}\);
Step (2) given any \(x_{k}\in X_{0}\), if \(x_{k}\in\bar{X}\), then the algorithm stops; otherwise go to Step (3);
Step (3) take \(x_{k}\notin\bar{X}\); we define \(x_{k+1}\) by the following expression:
where the sequence \(\lambda_{k}\in C^{0}\), \(\varepsilon_{k}\in(0, \varepsilon]\), \(\varepsilon>0\), and \(VN^{w}_{X_{0}}(\cdot)\) is the weak normal mapping to \(X_{0}\), f is a coercive Bregman function; go to Step (2).
Remark 3.1
The algorithm GPTM is actually a kind of exact proximal point algorithm, where the sequence \(\{\lambda_{k}\}\in C^{0}\) is called a sequence of scalarization parameters, a bounded exogenous sequence of positive real numbers \(\{\varepsilon_{k}\}\) is called a sequence of regularization parameters. For every \(x_{k}\notin\bar{X}\), we try to find a \(x_{k+1}\) such that \(0\in X^{*}\) belongs to the inclusion (3.1).
Next we will show the main results of this paper.
Theorem 3.1
Let \(X_{0}\subset X\) be nonempty, closed, and convex, \(T: X_{0} \rightarrow L(X, Y)\) be continuous and Cmonotone on \(X_{0}\), if \(\operatorname {dom}T\cap \operatorname {int}X_{0}\neq\emptyset\). The sequence \(\{x_{k}\}\) generated by the method (GPTM) is well defined.
Proof
Let \(x_{0}\in X_{0}\) be an initial point and suppose that the method (GPTM) reaches Step \((k)\). We then show that the next iterate \(x_{k+1}\) does exist. By the assumptions, \(T(\cdot)\) is continuous and Cmonotone on \(X_{0}\). By virtue of Proposition 2.1, we see that \(\lambda(T)\) is monotone and continuous on \(X_{0}\) for any \(\lambda\in C^{0}\). From Proposition 3.1, there exists a \(\bar{\lambda}\in C^{0}\) such that the mapping \(VN^{w}_{X_{0}}(\cdot){\bar{\lambda}}\) is a normal mapping on \(X_{0}\). Thus, by the assumption \(\operatorname {dom}T\cap \operatorname {int}X_{0}\neq\emptyset\) and the statement (a) of Lemma 2.2, one sees that, for any \(x\in X_{0}\), the mapping \((VN_{X_{0}}(x)+ T(x))\bar{\lambda}\) is maximal monotone. Without loss of generality, let \(\lambda_{k}=\bar{\lambda}\). Once again by the statement (b) of Lemma 2.2, we see that \(T(\cdot)\lambda_{k}+VN^{w}_{X_{0}}(\cdot) \lambda_{k}+ \varepsilon_{k} \nabla f(\cdot)\) is maximal monotone. Taking \(\varepsilon_{k} \nabla f\) as \(T_{1}\) in Lemma 2.3, it is easy to check that all assumptions are satisfied. It follows that \(R(T(\cdot)\lambda_{k}+VN^{w}_{X_{0}}(\cdot) \lambda_{k}+ \varepsilon_{k} \nabla f(\cdot))=X^{*}\). Then we see that, for any given \(\varepsilon_{k} \nabla f(x_{k})\in X^{*}\), there exists a \(x_{k+1}\) such that
The proof is complete. □
Theorem 3.2
Let the same assumptions as those in Theorem 3.1 hold. Further suppose that \(X_{0}^{\infty}\cap[T(X_{0})]^{w_{0}}_{C}=\{0\}\) and X̄ is nonempty and bounded. Then the sequence \(\{x_{k}\}\) generated by the method (GPTM) is bounded and
Proof
From the method (GPTM), we know that if the algorithm stops at some iteration, the point \(x_{k}\) will be a constant thereafter. Now we assume that the sequence \(\{x_{k}\}\) will not stop after a finite number of iterations. From Proposition 3.1, we know that there exist \(\lambda_{k}\in C^{0}\) and \(\varphi_{k+1}\in VN^{w}_{X_{0}}(x_{k+1})\) such that \(\lambda_{k}(\varphi_{k+1})\in N_{X_{0}}(x_{k+1})\). From the inclusion (3.1), one has
By the fact that \(\lambda_{k}(\varphi_{k+1})\in N_{X_{0}}(x_{k+1})\), we obtain
It follows that
On the other hand, from the assumption that X̄ is nonempty and bounded, without loss of generality, let \(x^{*}\in\bar{X}\). By virtue of Theorem 3.3 in [41], we know that \(x^{*}\) is also a solution of the problem (VIP\(_{\lambda_{k}}\)), where
It follows that
By the Cmonotonicity of T, one has
Combining (3.5) with (3.6), we obtain
By Property 2.1 of [22] (‘three points property’), we have
for any \(x\in X_{0}\). Hence from the inclusion (3.1), we know there exists \(\lambda_{k}(\varphi_{k+1})\in VN_{X_{0}}(x_{k+1})\) such that
Combining (3.7) with (3.8), we obtain
Since \(\lambda_{k}(\varphi_{k+1})\in N_{X_{0}}(x_{k+1})\), we have
It follows that
Clearly, we have \(\frac{1}{\varepsilon_{k}}\langle T(x_{k+1})\lambda _{k}, x_{k+1}x^{*}\rangle\geq0\). It follows that
Hence we see that the sequence \(B_{f}(x^{*}, x_{k})\) is decreasing as \(k\rightarrow\infty\). Taking \(\alpha= B_{f}(x^{*}, x_{0})\), we obtain the sequence \(\{x_{k}\}\subset S_{x^{*}, \alpha}\), which is a bounded set, by the assumption \(A_{1}\) of the Bregman function.
As for the sequence \(B_{f}(x^{*}, x_{k})\), it converges to a limit \(l^{*}\). From (3.12), we have
Summing up the above inequality, we obtain
The proof is complete. □
Theorem 3.3
Let the same assumptions as those in Theorem 3.2 hold. If we assume that the Bregman function f is uniformly totally convex, then any weak accumulation point of \(\{x_{k}\}\) is a solution of the problem (WVVI).
Proof
If there exists \(k_{0} \geq1\) such that \(x_{k_{0}+p}=x_{k_{0}}\), \(\forall p\geq1\), then it is clear that \(x_{k_{0}}\) is the unique weak cluster point of \(\{ x_{k}\}\) and it is also a solution of problem (WVVI). Suppose that the algorithm does not terminate finitely. Then, by Theorem 3.2, we see that \(\{x_{k}\}\) is bounded and it has some weak cluster points. Next we show that all of weak cluster points are solutions of problem (WVVI). Let x̂ be a weak cluster point of \(\{x_{k}\}\) and \(\{x_{k_{j}}\}\) be a subsequence of \(\{x_{k}\}\), which weakly converges to x̂. From the inequality (3.3), we know that
Hence by the assumption \(A_{3}\) of f, we obtain
Next we will show that \(\lim_{j\rightarrow+\infty} \Vert x_{k_{j}+1} x_{k_{j}}\Vert =0\). Suppose the contrary, without loss of generality we assume there exists a \(\beta>0\) such that
for all \(j\geq0\). Let D be a bounded set which contains the sequence \(\{x_{k_{j}}\}\). We have
From the assumption (3.17) and Property 2.2 in [22], we obtain
Since we assume f is uniformly totally convex, it follows that \(\inf_{x\in D}v_{f}(x, \beta)>0\). One has
Clearly, there is a contradiction between (3.20) and (3.15), hence we have
By the assumption \(A_{2}\) of the Bregman function f, one has
By the inclusion (3.1), one sees that there exist \(\lambda_{k_{j}}\in C^{0}\) and \(\varphi_{k_{j}+1}\in VN^{w}_{X_{0}}(x_{k_{j}+1})\), such that
It follows that
\(\{\lambda_{k_{j}}\}\subset C^{0}\), which is a weak^{∗}compact set. Without loss of generality, we assume the sequence \(w^{*}\lim_{j\rightarrow+\infty}\lambda_{k_{j}}=\bar{\lambda}\). From Proposition 3.1, for every j we have
By the demiclosedness of the graph \(T+N_{X_{0}}\) (see Proposition 2.3), we conclude that
Meanwhile, from Proposition 3.1, we know that there exists a \(\bar{\omega}\in VN^{w}_{X_{0}}(\bar{x})\bar{\lambda}\). By the definition of \(N_{X_{0}}(\bar{x})\), we know that
That is,
Thus
We conclude that x̄ is a solution of problem (WVVI). The proof is complete. □
Theorem 3.4
Let the same assumptions as those in Theorem 3.3 hold. If ∇f is a weak to weak continuous mapping, then the whole sequence \(\{x_{k}\}\) converges weakly to a solution of problem (WVVI).
Proof
In this part we will show x̄ is the only solution of (WVVI). Suppose the contrary, assume that both x̂ and x̄ are weak cluster points of \(\{x_{n}\}\)
and x̂ and x̄ are also solutions of (WVVI). From the proof of Theorem 3.2, we see that the sequences \(B_{f}(\hat {x}, x_{k})\) and \(B_{f} (\bar{x}, x_{k})\) are bounded and let \(l_{1}\) and \(l_{2}\) be their limits, then
Let \(\lim_{k\rightarrow+\infty}\langle\nabla f(x_{k}), \bar{x}\hat{x}\rangle=l\). Taking \(k=k_{j}\) in (3.25) and by virtue of the weak to weak continuity of \(\nabla f(\cdot)\) we obtain
Once again taking \(k=k_{i}\) in (3.25) and by virtue of the weak to weak continuity of \(\nabla f(\cdot)\) we obtain
Combining the equality (3.26) with the equality (3.27), one has
From the strict convexity of f, we have \(\bar{x}=\hat{x}\), which means x̄ is the only solution of (WVVI). The proof is complete. □
Conclusions
In this paper, we proposed a class of generalized proximaltype method by virtue of the Bregman distance functions to solve weak vector variational inequality problems in Banach spaces. We carried out a convergence analysis on the method and proved weak convergence of the generated sequence to a solution of the weak vector variational inequality problems under some mild conditions. Our results extended some wellknown results to more general cases.
References
Martinet, B: Regularisation d’inéquations variationelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 2, 154159 (1970)
Rockafellar, RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877898 (1976)
Auslender, A, Haddou, M: An interior proximal point method for convex linearly constrained problems and its extension to variational inequalities. Math. Program. 71, 77100 (1995)
Güler, O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403419 (1991)
Kamimura, S, Takahashi, W: Strong convergence of a proximaltype algorithm in a Banach space. SIAM J. Optim. 13(3), 938945 (2003)
Pathak, HK, Cho, YJ: Strong convergence of a proximaltype algorithm for an occasionally pseudomonotone operator in Banach spaces. Fixed Point Theory Appl. 2012, 190 (2012)
Pennanen, T: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27(1), 170191 (2002)
Solodov, MV, Svaiter, BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program., Ser. A 87, 189202 (2000)
Bonnel, H, Iusem, AN, Svaiter, BF: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953970 (2005)
Ceng, LC, Yao, JC: Approximate proximal methods in vector optimization. Eur. J. Oper. Res. 181(1), 119 (2007)
Chen, Z, Huang, XX, Yang, XQ: Generalized proximal point algorithms for multiobjective optimization problems. Appl. Anal. 90, 935949 (2011)
Chen, Z, Zhao, KQ: A proximaltype method for convex vector optimization problem in Banach spaces. Numer. Funct. Anal. Optim. 30, 7081 (2009)
Chen, Z, Huang, HQ, Zhao, KQ: Approximate generalized proximaltype method for convex vector optimization problem in Banach spaces. Comput. Math. Appl. 57, 11961203 (2009)
Chen, Z: Asymptotic analysis in convex composite multiobjective optimization problems. J. Glob. Optim. 55, 507520 (2013)
Bregman, LM: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200217 (1967). doi:10.1016/00415553(67)900407
Censor, Y, Lent, A: An iterative rowaction method for interval convex programming. J. Optim. Theory Appl. 34, 321353 (1981)
Bauschke, HH, Borwein, JM: Legendre function and the method of random Bregman functions. J. Convex Anal. 4, 2767 (1997)
Censor, Y, Zenios, SA: The proximal minimization algorithm with Dfunction. J. Optim. Theory Appl. 73, 451464 (1992)
Chen, G, Teboulle, M: Convergence analysis of a proximallike minimization algorithm using Bregman function. SIAM J. Optim. 3(3), 538543 (1993)
Kiwiel, KC: Proximal minimization methods with generalized Bregman functions. SIAM J. Control Optim. 35, 326349 (1997)
Burachik, RS, Iusem, AN: A generalized proximal point algorithm for the variational inequality problem in a Hilbert space. SIAM J. Optim. 8(1), 197216 (1998)
Burachik, RS, Scheimberg, S: A proximal point algorithm for the variational inequality problem in Banach space. SIAM J. Control Optim. 39(5), 16331649 (2001)
Giannessi, F: Theorems of alternative, quadratic programs and complementarity problems. In: Cottle, RW, Giannessi, F, Lions, JL (eds.) Variational Inequality and Complementarity Problems. Wiley, New York (1980)
Chen, GY, Yang, XQ: The vector complementary problem and its equivalences with vector minimal element in ordered spaces. J. Math. Anal. Appl. 153, 136158 (1990)
Chen, GY: Existence of solutions for a vector variational inequality: an extension of the HartmanStampacchia theorem. J. Optim. Theory Appl. 74, 445456 (1992)
Chen, GY, Huang, XX, Yang, XQ: Vector Optimization, SetValued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)
Chen, Z: Asymptotic analysis for proximaltype methods in vector variational inequality problems. Oper. Res. Lett. 43, 226230 (2015)
Allgower, EL, Böhmer, K, Potra, FA, Rheinboldt, WC: A mesh independence principle for operator equations and their discretizations. SIAM J. Numer. Anal. 23, 160169 (1986)
Heinkenschloss, M: Mesh independence for nonlinear least squares problems with norm constraints. SIAM J. Optim. 3, 81117 (1993)
Laumen, M: Newton’s mesh independence principle for a class of optimal shape design problems. SIAM J. Control Optim. 37, 11421168 (1999)
Huang, XX, Fang, YP, Yang, XQ: Characterizing the nonemptiness and compactness of the solution set of a vector variational inequality by scalarization. J. Optim. Theory Appl. 162(2), 548558 (2014)
Facchinei, F, Pang, JS: FiniteDimensional Variational Inequalities and Complementarity Problems, Volume I, II. Springer Series in Operations Research. Springer, Berlin (2004)
Hu, R, Fang, YP: On the nonemptiness and compactness of the solution sets for vector variational inequalities. Optimization 59, 11071116 (2010)
Rockafellar, RT: Convex Analysis. Princeton University Press, Princeton (1970)
Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 7588 (1970)
Auslender, A: Optimisation. Methodes Numeriques. Masson, Paris (1976)
Brezis, H, Haraux, A: Image d’une somme d’operateurs monotones et applications. Isr. J. Math. 23, 165186 (1976)
Pascali, D, Sburlan, S: Nonlinear Mappings of Monotone Type. Nijhoff, Dordrecht (1978)
Butnariu, D, Iusem, AN: Local moduli of convexity and their applications to finding almost common points of measurable families of operators. In: Censor, Y, Reich, S (eds.) Recent Developments in Optimization Theory and Nonlinear Analysis. Contemp. Math., vol. 204, pp. 6191. AMS, Providence (1997)
Attouch, H, Buttazzo, G, Michaille, G: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MPS/SIAM, Philadelphia (2006)
Fan, JH, Jing, Y, Zhong, RY: Nonemptiness and boundedness of solution sets for vector variational inequalities via topological method. J. Glob. Optim. 63, 181193 (2015). doi:10.1007/s1089801502792
Acknowledgements
This work is partially supported by the National Science Foundation of China (No. 11001289, 71471122). The authors thank the anonymous referees for their helpful suggestions, which have contributed to a major improvement of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Pu, Lc., Wang, Xy. & Chen, Z. Generalized proximaltype methods for weak vector variational inequality problems in Banach spaces. Fixed Point Theory Appl 2015, 191 (2015). https://doi.org/10.1186/s1366301504400
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366301504400
MSC
 90C29
 65K10
 49M05
Keywords
 vector variational inequality problems
 proximaltype method
 Banach spaces
 Bregman functions
 weak normal mapping
 convergence analysis