A projection method for bilevel variational inequalities
© Anh et al.; licensee Springer. 2014
Received: 19 October 2013
Accepted: 8 May 2014
Published: 22 May 2014
A fixed point iteration algorithm is introduced to solve bilevel monotone variational inequalities. The algorithm uses simple projection sequences. Strong convergence of the iteration sequences generated by the algorithm to the solution is guaranteed under some assumptions in a real Hilbert space.
Keywordsbilevel variational inequalities pseudomonotonicity strongly monotone Lipschitz continuous global convergence
and . We denote the solution set of problem (BVI) by Ω.
Bilevel variational inequalities are special classes of quasivariational inequalities (see [1–4]) and of equilibrium with equilibrium constraints considered in . However, they cover some classes of mathematical programs with equilibrium constraints (see ), bilevel minimization problems (see ), variational inequalities (see [8–13]), minimum-norm problems of the solution set of variational inequalities (see [14, 15]), bilevel convex programming models (see ) and bilevel linear programming in .
They showed that under certain conditions over parameters, the sequence converges strongly to .
Under assumptions that F is strongly monotone and Lipschitz continuous, G is pseudomonotone and Lipschitz continuous on C, the sequences of parameters were chosen appropriately. They showed that two iterative sequences and converged to the same point which is a solution of problem (BVI). However, at each iteration of the outer loop, the scheme requires computing an approximation solution to a variational inequality problem.
There exist some other solution methods for bilevel variational inequalities when the cost operator has some monotonicity (see [16, 19–21]). In all of these methods, solving auxiliary variational inequalities is required. In order to avoid this requirement, we combine the projected gradient method in  for solving variational inequalities and the fixed point property that is a solution to problem if and only if it is a fixed point of the mapping , where . Then, the strong convergence of proposed sequences is considered in a real Hilbert space.
In this paper, we are interested in finding a solution to bilevel variational inequalities (BVI), where the operators F and G satisfy the following usual conditions:
(A1) G is η-inverse strongly monotone on ℋ and F is β-strongly monotone on C.
(A2) F is L-Lipschitz continuous on C.
(A3) The solution set Ω of problem (BVI) is nonempty.
The purpose of this paper is to propose an algorithm for directly solving bilevel pseudomonotone variational inequalities by using the projected gradient method and fixed point techniques.
The rest of this paper is divided into two sections. In Section 2, we recall some properties for monotonicity, the metric projection onto a closed convex set and introduce in detail a new algorithm for solving problem (BVI). The third section is devoted to the convergence analysis for the algorithm.
We list some well-known definitions and the projection under the Euclidean norm which will be used in our analysis.
- (i)γ-strongly monotone on C if for each ,
- (ii)η-inverse strongly monotone on C if for each ,
- (iii)Lipschitz continuous with constant (shortly L-Lipschitz continuous) on C if for each ,
If and , then φ is called nonexpansive on C.
We know that the projection has the following well-known basic properties.
To prove the main theorem of this paper, we need the following lemma.
Lemma 2.3 (see )
Lemma 2.4 (see )
Let ℋ be a real Hilbert space, C be a nonempty closed and convex subset of ℋ and be a nonexpansive mapping. Then (I is the identity operator on ℋ) is demiclosed at , i.e., for any sequence in C such that and , we have .
Lemma 2.5 (see )
Now we are in a position to describe an algorithm for problem (BVI). The proposed algorithm can be considered as a combination of the projected gradient and fixed point methods. Roughly speaking the algorithm consists of two steps. First, we use the well-known projected gradient method for solving the variational inequalities (), where and . The method generates a sequence converging strongly to the unique solution of problem under assumptions that G is L-Lipschitz continuous and α-strongly monotone on C with the step-size . Next, we use the Banach contraction-mapping fixed-point principle for finding the unique fixed point of the contraction-mapping , where F is β-strongly monotone and L-Lipschitz continuous, I is the identity mapping, and . The algorithm is presented in detail as follows.
Algorithm 2.6 (Projection algorithm for solving (BVI))
Update , and go to Step 1.
3 Convergence results
In this section, we state and prove our main results.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Let two mappings and satisfy assumptions (A1)-(A3). Then the sequences and in Algorithm 2.6 converge strongly to the same point .
It follows from (3.1) that the mapping is nonexpansive on ℋ. Using Lemma 2.4, (3.3) and , we obtain , which implies . Now we will prove that .
Let , and hence the sequence converges strongly to . This implies that the sequence also converges strongly to .
Applying Lemma 2.5, . Combining this and the fact that the sequence converges strongly to , the sequence also converges strongly to the unique solution to problem (BVI). □
Now we consider the special case for all . It is easy to see that F is Lipschitz continuous with constant and strongly monotone with constant on ℋ. Problem (BVI) becomes the minimum-norm problems of the solution set of the variational inequalities.
Then the sequences and converge strongly to the same point .
The authors are very grateful to the anonymous referees for their really helpful and constructive comments that helped us very much in improving the paper.
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