A New Method for Solving Monotone Generalized Variational Inequalities
© Pham Ngoc Anh and Jong Kyu Kim. 2010
Received: 11 May 2010
Accepted: 4 October 2010
Published: 12 October 2010
We suggest new dual algorithms and iterative methods for solving monotone generalized variational inequalities. Instead of working on the primal space, this method performs a dual step on the dual space by using the dual gap function. Under the suitable conditions, we prove the convergence of the proposed algorithms and estimate their complexity to reach an -solution. Some preliminary computational results are reported.
In recent years, this generalized variational inequalities become an attractive field for many researchers and have many important applications in electricity markets, transportations, economics, and nonlinear analysis (see [1–9]).
It is well known that the interior quadratic and dual technique are powerfull tools for analyzing and solving the optimization problems (see [10–16]). Recently these techniques have been used to develop proximal iterative algorithm for variational inequalities (see [17–22]).
In addition Nesterov  introduced a dual extrapolation method for solving variational inequalities. Instead of working on the primal space, this method performs a dual step on the dual space.
In this paper we extend results in  to the generalized variational inequality problem (GVI) in the dual space. In the first approach, a gap function is constructed such that , for all and if and only if solves (GVI). Namely, we first develop a convergent algorithm for (GVI) with being monotone function satisfying a certain Lipschitz type condition on . Next, in order to avoid the Lipschitz condition we will show how to find a regularization parameter at every iteration such that the sequence converges to a solution of (GVI).
The remaining part of the paper is organized as follows. In Section 2, we present two convergent algorithms for monotone and generalized variational inequality problems with Lipschitzian condition and without Lipschitzian condition. Section 3 deals with some preliminary results of the proposed methods.
First, let us recall the well-known concepts of monotonicity that will be used in the sequel (see ).
Note that when is differentiable on some open set containing , then, since is lower semicontinuous proper convex, the generalized variational inequality (GVI) is equivalent to the following variational inequalities (see [25, 26]):
Throughout this paper, we assume that:
The following lemma gives two basic properties of the dual gap function (2.7) whose proof can be found, for instance, in .
Under assumptions (A1)–(A4), the following properties hold.
The following lemma gives a tool for the next discussion.
which proves (2.16).
3. Dual Algorithms
Now, we are going to build the dual interior proximal step for solving (GVI). The main idea is to construct a sequence such that the sequence tends to 0 as . By virtue of Lemma 2.5, we can check whether is an -solution to (GVI) or not.
which proves (3.2).
Then the inequality (3.3) is deduced from this inequality and (3.6).
One has the following.
Now, we prove the convergence of Algorithm 3.2 and estimate its complexity.
where is the final output defined by the sequence in Algorithm 3.2. As a consequence, the sequence converges to 0 and the number of iterations to reach an -solution is , where denotes the largest integer such that .
then the algorithm can be modified to ensure that it still converges. The variant of Algorithm 3.2 is presented as Algorithm 3.4 below.
One has the following.
The next theorem shows the convergence of Algorithm 3.4.
4. Illustrative Example and Numerical Results
For this class of problem (GVI) we have the following results.
Then (i) and (ii) easily follow.
Using the Lipschitz condition, it is not difficult to obtain (iii).
where the components of the are defined by: , with randomly chosen in and the components of are randomly chosen in . The function is given by Bnouhachem . Under these assumptions, it can be proved that is continuous and monotone on .
The authors would like to thank the referees for their useful comments, remarks and suggestions. This work was completed while the first author was staying at Kyungnam University for the NRF Postdoctoral Fellowship for Foreign Researchers. And the second author was supported by Kyungnam University Research Fund, 2010.
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