- Research Article
- Open Access

# A New Projection Algorithm for Generalized Variational Inequality

- Changjie Fang
^{1, 2}Email author and - Yiran He
^{1}

**2010**:182576

https://doi.org/10.1155/2010/182576

© C. Fang and Y. He. 2010

**Received:**26 October 2009**Accepted:**20 December 2009**Published:**24 January 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.

## Keywords

- Variational Inequality
- Multivalued Mapping
- Variational Inequality Problem
- Projection Algorithm
- Proximal Point Algorithm

## 1. Introduction

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 [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:

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.

Algorithm 2.2.

Choose and three parameters , and Set

Step 1.

If for some , stop; else take arbitrarily .

Step 2.

Step 3.

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.

So . Let in (2.4), we have . This contradiction completes the proof.

Lemma 2.6.

Proof.

See [15, Lemma ].

Lemma 2.7.

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.

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.

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 .

## 4. Numerical Experiments

Example 4.1.

Algorithm 2.2 | [15, Algorithm ] | |||
---|---|---|---|---|

It. (num.) | CPU (sec.) | It. (num.) | CPU (sec.) | |

55 | 0.625 | 74 | 0.984375 | |

39 | 0.546875 | 51 | 0.75 | |

23 | 0.4375 | 27 | 0.5 |

Example 4.2.

Algorithm 2.2 | [15, Algorithm ] | ||||
---|---|---|---|---|---|

Initial point | It. (num.) | CPU (sec.) | It. (num.) | CPU (sec.) | |

(0,0,0,1) | 53 | 0.75 | 61 | 0.90625 | |

(0,0,1,0) | 47 | 0.625 | 79 | 1.28125 | |

(0.5,0,0.5,0) | 42 | 0.53125 | 76 | 1.28125 | |

(0,0,0,1) | 42 | 0.625 | 43 | 0.671875 | |

(0,0,1,0) | 35 | 0.53125 | 56 | 0.921875 | |

(0.5,0,0.5,0) | 31 | 0.5 | 53 | 0.890625 |

Example 4.1.

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.

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.

## Declarations

### 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).

## Authors’ Affiliations

## References

- Allevi E, Gnudi A, Konnov IV: The proximal point method for nonmonotone variational inequalities.
*Mathematical Methods of Operations Research*2006, 63(3):553–565. 10.1007/s00186-005-0052-2MathSciNetView ArticleMATHGoogle Scholar - Auslender A, Teboulle M: Lagrangian duality and related multiplier methods for variational inequality problems.
*SIAM Journal on Optimization*2000, 10(4):1097–1115. 10.1137/S1052623499352656MathSciNetView ArticleMATHGoogle Scholar - Bao TQ, Khanh PQ: A projection-type algorithm for pseudomonotone nonlipschitzian multivalued variational inequalities. In
*Generalized Convexity, Generalized Monotonicity and Applications, Nonconvex Optimization and Its Applications*.*Volume 77*. Springer, New York, NY, USA; 2005:113–129. 10.1007/0-387-23639-2_6View ArticleGoogle Scholar - Ceng LC, Mastroeni G, Yao JC: An inexact proximal-type method for the generalized variational inequality in Banach spaces.
*Journal of Inequalities and Applications*2007, 2007:-14.Google Scholar - Fang SC, Peterson EL: Generalized variational inequalities.
*Journal of Optimization Theory and Applications*1982, 38(3):363–383. 10.1007/BF00935344MathSciNetView ArticleMATHGoogle Scholar - Fukushima M: The primal Douglas-Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem.
*Mathematical Programming*1996, 72(1):1–15. 10.1007/BF02592328MathSciNetView ArticleMATHGoogle Scholar - He Y: Stable pseudomonotone variational inequality in reflexive Banach spaces.
*Journal of Mathematical Analysis and Applications*2007, 330(1):352–363. 10.1016/j.jmaa.2006.07.063MathSciNetView ArticleMATHGoogle Scholar - Saigal R: Extension of the generalized complementarity problem.
*Mathematics of Operations Research*1976, 1(3):260–266. 10.1287/moor.1.3.260MathSciNetView ArticleMATHGoogle Scholar - Salmon G, Strodiot J-J, Nguyen VH: A bundle method for solving variational inequalities.
*SIAM Journal on Optimization*2003, 14(3):869–893.MathSciNetView ArticleMATHGoogle Scholar - Rockafellar RT: Monotone operators and the proximal point algorithm.
*SIAM Journal on Control and Optimization*1976, 14(5):877–898. 10.1137/0314056MathSciNetView ArticleMATHGoogle Scholar - Konnov IV: On the rate of convergence of combined relaxation methods.
*Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika*1993, (12):89–92.Google Scholar - Konnov IV:
*Combined Relaxation Methods for Variational Inequalities, Lecture Notes in Economics and Mathematical Systems*.*Volume 495*. Springer, Berlin, Germany; 2001:xii+181.View ArticleGoogle Scholar - Konnov IV: Combined relaxation methods for generalized monotone variational inequalities. In
*Generalized Convexity and Related Topics, Lecture Notes in Econom. and Math. Systems*.*Volume 583*. Springer, Berlin, Germany; 2007:3–31.View ArticleGoogle Scholar - He Y: A new double projection algorithm for variational inequalities.
*Journal of Computational and Applied Mathematics*2006, 185(1):166–173. 10.1016/j.cam.2005.01.031MathSciNetView ArticleMATHGoogle Scholar - Li F, He Y: An algorithm for generalized variational inequality with pseudomonotone mapping.
*Journal of Computational and Applied Mathematics*2009, 228(1):212–218. 10.1016/j.cam.2008.09.014MathSciNetView ArticleMATHGoogle Scholar - Solodov MV, Svaiter BF: A new projection method for variational inequality problems.
*SIAM Journal on Control and Optimization*1999, 37(3):765–776. 10.1137/S0363012997317475MathSciNetView ArticleMATHGoogle Scholar - Facchinei F, Pang JS:
*Finite-Dimensional Variational Inequalities and Complementary Problems*. Springer, New York, NY, USA; 2003.MATHGoogle Scholar - Karamardian S: Complementarity problems over cones with monotone and pseudomonotone maps.
*Journal of Optimization Theory and Applications*1976, 18(4):445–454. 10.1007/BF00932654MathSciNetView ArticleMATHGoogle Scholar - Aubin J-P, Ekeland I:
*Applied Nonlinear Analysis, Pure and Applied Mathematics*. John Wiley & Sons, New York, NY, USA; 1984:xi+518.MATHGoogle Scholar

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