- Research Article
- Open Access
A New Projection Algorithm for Generalized Variational Inequality
© C. Fang and Y. He. 2010
- Received: 26 October 2009
- Accepted: 20 December 2009
- Published: 24 January 2010
We propose a new projection algorithm for generalized variational inequality with multivalued mapping. Our method is proven to be globally convergent to a solution of the variational inequality problem, provided that the multivalued mapping is continuous and pseudomonotone with nonempty compact convex values. Preliminary computational experience is also reported.
- Variational Inequality
- Multivalued Mapping
- Variational Inequality Problem
- Projection Algorithm
- Proximal Point Algorithm
We consider the following generalized variational inequality. To find and such that
where is a nonempty closed convex set in , is a multivalued mapping from into with nonempty values, and and denote the inner product and norm in , respectively.
Theory and algorithm of generalized variational inequality have been much studied in the literature [1–9]. Various algorithms for computing the solution of (1.1) are proposed. The well-known proximal point algorithm  requires the multivalued mapping to be monotone. Relaxing the monotonicity assumption,  proved if the set is a box and is order monotone, then the proximal point algorithm still applies for problem (1.1). Assume that is pseudomonotone, and  described a combined relaxation method for solving (1.1); see also [12, 13]. Projection-type algorithms have been extensively studied in the literature; see [14–17] and the references therein. Recently,  proposes a projection algorithm for generalized variational inequality with pseudomonotone mapping. In , choosing needs solving a single-valued variational inequality and hence is computationally expensive; see expression (2.1) in . In this paper, we introduce a different projection algorithm for generalized variational inequality. In our method, can be taken arbitrarily. Moreover, the main difference of our method from that of  is the procedure of Armijo-type linesearch; see expression (2.2) in  and expression (2.2) in the next section.
Let be the solution set of (1.1), that is, those points satisfying (1.1). Throughout this paper, we assume that the solution set of problem (1.1) is nonempty and is continuous on with nonempty compact convex values satisfying the following property:
Property (1.2) holds if is pseudomonotone on in the sense of Karamardian . In particular, if is monotone, then (1.2) holds.
The organization of this paper is as follows. In the next section, we recall the definition of continuous multivalued mapping, present the algorithm details, and prove the preliminary result for convergence analysis in Section 3. Numerical results are reported in the last section.
Let us recall the definition of continuous multivalued mapping. is said to be upper semicontinuous at if for every open set containing , there is an open set containing such that for all . F is said to be lower semicontinuous at , if we give any sequence converging to and any , there exists a sequence that converges to . is said to be continuous at if it is both upper semicontinuous and lower semicontinuous at . If is single valued, then both upper semicontinuity and lower semicontinuity reduce to the continuity of .
Let denote the projector onto and let be a parameter.
Choose and three parameters , and Set
If for some , stop; else take arbitrarily .
where . Set .
Let and go to Step 1.
Since has compact convex values, has closed convex values. Therefore, in Step 2 is uniquely determined by .
If is a single-valued mapping, the Armijo-type linesearch procedure (2.2) becomes that of Algorithm 2.2 in .
We show that Algorithm 2.2 is well defined and implementable.
If is not a solution of problem (1.1), then there exists a nonnegative integer satisfying (2.2).
So . Let in (2.4), we have . This contradiction completes the proof.
See [15, Lemma ].
where denotes the distance from to .
See [14, Lemma ].
Let solve the variational inequality (1.1) and let the function be defined by (2.3). Then and . In particular, if then
where the second inequality follows from assumption (1.2) and . Thus is verified.
If is continuous with nonempty compact convex values on and condition (1.2) holds, then either Algorithm 2.2 terminates in a finite number of iterations or generates an infinite sequence converging to a solution of (1.1).
By the boundedness of , there exists a convergent subsequence converging to .
If is a solution of problem (1.1), we show next that the whole sequence converges to . Replacing by in the preceding argument, we obtain that the sequence is nonincreasing and hence converges. Since is an accumulation point of , some subsequence of converges to zero. This shows that the whole sequence converges to zero, hence .
with being continuous. Therefore, is bounded and so is .
By the boundedness of , we obtain that Since is continuous and the sequences and are bounded, there exists an accumulation point of such that . This implies that solves the variational inequality (1.1). Similar to the preceding proof, we obtain that .
[15, Algorithm ]
[15, Algorithm ]
Then the set and the mapping satisfy the assumptions of Theorem 3.1 and (0,0,1) is a solution of the generalized variational inequality. Example 4.1 is tested in . We choose , and for our algorithm; , and for Algorithm 1 in . We use as the initial point.
Then the set and the mapping satisfy the assumptions of Theorem 3.1 and (1,0,0,0) is a solution of the generalized variational inequality. We choose , and for the two algorithms.
This work is partially supported by the National Natural Science Foundation of China (no. 10701059), by the Sichuan Youth Science and Technology Foundation (no. 06ZQ026-013), and by Natural Science Foundation Projection of CQ CSTC (no. 2008BB7415).
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