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A New Projection Algorithm for Generalized Variational Inequality
Journal of Inequalities and Applications volume 2010, Article number: 182576 (2010)
Abstract
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.
1. Introduction
We consider the following generalized variational inequality. To find and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ1_HTML.gif)
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 [10] requires the multivalued mapping to be monotone. Relaxing the monotonicity assumption, [1] 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 [11] 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, [15] proposes a projection algorithm for generalized variational inequality with pseudomonotone mapping. In [15], choosing
needs solving a single-valued variational inequality and hence is computationally expensive; see expression (2.1) in [15]. 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 [15] is the procedure of Armijo-type linesearch; see expression (2.2) in [15] 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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ2_HTML.gif)
Property (1.2) holds if is pseudomonotone on
in the sense of Karamardian [18]. 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.
2. Algorithms
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.
Proposition 2.1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_IEq47_HTML.gif)
and solve problem (1.1) if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ3_HTML.gif)
Algorithm 2.2.
Choose and three parameters
, and
Set
Step 1.
If for some
, stop; else take arbitrarily
.
Step 2.
Let be the smallest nonnegative integer satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ4_HTML.gif)
where . Set
.
Step 3.
Compute where
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ5_HTML.gif)
Let and go to Step 1.
Remark 2.3.
Since has compact convex values,
has closed convex values. Therefore,
in Step 2 is uniquely determined by
.
Remark 2.4.
If is a single-valued mapping, the Armijo-type linesearch procedure (2.2) becomes that of Algorithm 2.2 in [14].
We show that Algorithm 2.2 is well defined and implementable.
Proposition 2.5.
If is not a solution of problem (1.1), then there exists a nonnegative integer
satisfying (2.2).
Proof.
Suppose that for all , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ6_HTML.gif)
where . Since
is lower semicontinuous,
, and
as
, for each
, there is
such that
. Since
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ7_HTML.gif)
So . Let
in (2.4), we have
. This contradiction completes the proof.
Lemma 2.6.
For every and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ8_HTML.gif)
Proof.
See [15, Lemma ].
Lemma 2.7.
Let be a closed convex set in
,
a real-valued function on
, and
the set
. If
is nonempty and
is Lipschitz continuous on
with modulus
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ9_HTML.gif)
where denotes the distance from
to
.
Proof.
See [14, Lemma ].
Lemma 2.8.
Let solve the variational inequality (1.1) and let the function
be defined by (2.3). Then
and
. In particular, if
then
Proof.
It follows from (2.3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ10_HTML.gif)
where the first inequality follows from (2.2) and the last one follows from Lemma 2.6 and . If
, then
because
. It remains to be proved that
. Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ11_HTML.gif)
on the other hand, assumption (1.2) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ12_HTML.gif)
Adding the last two expressions, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ13_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ14_HTML.gif)
where the second inequality follows from assumption (1.2) and . Thus
is verified.
3. Main Results
Theorem 3.1.
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).
Proof.
Let be a solution of the variational inequality problem. By Lemma 2.8,
. We assume that Algorithm 2.2 generates an infinite sequence
. In particular,
for every
. By Step 3, it follows from Lemma
in [14] that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ15_HTML.gif)
where the last inequality is due to . It follows that the squence
is nonincreasing, and hence is a convergent sequence. Therefore,
is bounded and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ16_HTML.gif)
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
.
Suppose now that is not a solution of problem (1.1). We show first that
in Algorithm 2.2 cannot tend to
. Since
is continuous with compact values, Proposition
in [19] implies that
is a bounded set, and so the sequence
is bounded. Therefore, there exists a subsequence
converging to
. Since
is upper semicontinuous with compact values, Proposition
in [19] implies that
is closed, and so
. By the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ17_HTML.gif)
If , then
. The lower continuity of
, in turn, implies the existence of
such that
converges to
. Since
,
, and
. Therefore
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ18_HTML.gif)
Letting , we obtain the contradiction
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ19_HTML.gif)
with being continuous. Therefore,
is bounded and so is
.
It follows from (2.3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ20_HTML.gif)
Since and
are bounded, we have the sequence
and hence the sequence
is bounded. Thus, for some
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ21_HTML.gif)
Therefore, each function is Lipschitz continuous on
with modulus
. Noting that
and applying Lemma 2.7, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ22_HTML.gif)
It follows from (3.8) and Lemma 2.8 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ23_HTML.gif)
Then (3.2) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ24_HTML.gif)
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
.
4. Numerical Experiments
In this section, we present some numerical experiments for the proposed algorithm. The MATLAB codes are run on a PC (with CPU Intel P-T2390) under MATLAB Version 7.0.1.24704(R14) Service Pack 1. We compare the performance of our Algorithm 2.2 and [15, Algorithm 1]. In the Tables 1 and 2, "It." denotes number of iteration, and "CPU" denotes the CPU time in seconds. The tolerance means when
the procedure stops.
Example 4.1.
Let ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ25_HTML.gif)
and let be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ26_HTML.gif)
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 [15]. We choose
, and
for our algorithm;
, and
for Algorithm 1 in [15]. We use
as the initial point.
Example 4.2.
Let ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ27_HTML.gif)
and be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F182576/MediaObjects/13660_2009_Article_2075_Equ28_HTML.gif)
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.
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Acknowledgments
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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Fang, C., He, Y. A New Projection Algorithm for Generalized Variational Inequality. J Inequal Appl 2010, 182576 (2010). https://doi.org/10.1155/2010/182576
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DOI: https://doi.org/10.1155/2010/182576