# A New Projection Algorithm for Generalized Variational Inequality

- 879 Downloads

## 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 Open image in new window and Open image in new window such that

where Open image in new window is a nonempty closed convex set in Open image in new window , Open image in new window is a multivalued mapping from Open image in new window into Open image in new window with nonempty values, and Open image in new window and Open image in new window denote the inner product and norm in Open image in new window , respectively.

Theory and algorithm of generalized variational inequality have been much studied in the literature [1, 2, 3, 4, 5, 6, 7, 8, 9]. Various algorithms for computing the solution of (1.1) are proposed. The well-known proximal point algorithm [10] requires the multivalued mapping Open image in new window to be monotone. Relaxing the monotonicity assumption, [1] proved if the set Open image in new window is a box and Open image in new window is order monotone, then the proximal point algorithm still applies for problem (1.1). Assume that Open image in new window 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, 15, 16, 17] and the references therein. Recently, [15] proposes a projection algorithm for generalized variational inequality with pseudomonotone mapping. In [15], choosing Open image in new window 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, Open image in new window 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 Open image in new window be the solution set of (1.1), that is, those points Open image in new window satisfying (1.1). Throughout this paper, we assume that the solution set Open image in new window of problem (1.1) is nonempty and Open image in new window is continuous on Open image in new window with nonempty compact convex values satisfying the following property:

Property (1.2) holds if Open image in new window is pseudomonotone on Open image in new window in the sense of Karamardian [18]. In particular, if Open image in new window 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. Open image in new window is said to be upper semicontinuous at Open image in new window if for every open set Open image in new window containing Open image in new window , there is an open set Open image in new window containing Open image in new window such that Open image in new window for all Open image in new window . F is said to be lower semicontinuous at Open image in new window , if we give any sequence Open image in new window converging to Open image in new window and any Open image in new window , there exists a sequence Open image in new window that converges to Open image in new window . Open image in new window is said to be continuous at Open image in new window if it is both upper semicontinuous and lower semicontinuous at Open image in new window . If Open image in new window is single valued, then both upper semicontinuity and lower semicontinuity reduce to the continuity of Open image in new window .

Let Open image in new window denote the projector onto Open image in new window and let Open image in new window be a parameter.

Proposition 2.1.

Algorithm 2.2.

Choose Open image in new window and three parameters Open image in new window , and Open image in new window Set Open image in new window

Step 1.

If Open image in new window for some Open image in new window , stop; else take arbitrarily Open image in new window .

Step 2.

where Open image in new window . Set Open image in new window .

Step 3.

Let Open image in new window and go to Step 1.

Remark 2.3.

Since Open image in new window has compact convex values, Open image in new window has closed convex values. Therefore, Open image in new window in Step 2 is uniquely determined by Open image in new window .

Remark 2.4.

If Open image in new window 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 Open image in new window is not a solution of problem (1.1), then there exists a nonnegative integer Open image in new window satisfying (2.2).

Proof.

So Open image in new window . Let Open image in new window in (2.4), we have Open image in new window . This contradiction completes the proof.

Lemma 2.6.

Proof.

See [15, Lemma Open image in new window ].

Lemma 2.7.

where Open image in new window denotes the distance from Open image in new window to Open image in new window .

Proof.

See [14, Lemma Open image in new window ].

Lemma 2.8.

Let Open image in new window solve the variational inequality (1.1) and let the function Open image in new window be defined by (2.3). Then Open image in new window and Open image in new window . In particular, if Open image in new window then Open image in new window

Proof.

where the second inequality follows from assumption (1.2) and Open image in new window . Thus Open image in new window is verified.

## 3. Main Results

Theorem 3.1.

If Open image in new window is continuous with nonempty compact convex values on Open image in new window and condition (1.2) holds, then either Algorithm 2.2 terminates in a finite number of iterations or generates an infinite sequence Open image in new window converging to a solution of (1.1).

Proof.

By the boundedness of Open image in new window , there exists a convergent subsequence Open image in new window converging to Open image in new window .

If Open image in new window is a solution of problem (1.1), we show next that the whole sequence Open image in new window converges to Open image in new window . Replacing Open image in new window by Open image in new window in the preceding argument, we obtain that the sequence Open image in new window is nonincreasing and hence converges. Since Open image in new window is an accumulation point of Open image in new window , some subsequence of Open image in new window converges to zero. This shows that the whole sequence Open image in new window converges to zero, hence Open image in new window .

with Open image in new window being continuous. Therefore, Open image in new window is bounded and so is Open image in new window .

By the boundedness of Open image in new window , we obtain that Open image in new window Since Open image in new window is continuous and the sequences Open image in new window and Open image in new window are bounded, there exists an accumulation point Open image in new window of Open image in new window such that Open image in new window . This implies that Open image in new window solves the variational inequality (1.1). Similar to the preceding proof, we obtain that Open image in new window .

## 4. Numerical Experiments

Example 4.1.

Algorithm 2.2 | [15, Algorithm Open image in new window ] | |||
---|---|---|---|---|

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 Open image in new window ] | ||||
---|---|---|---|---|---|

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 Open image in new window and the mapping Open image in new window 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 Open image in new window , and Open image in new window for our algorithm; Open image in new window , and Open image in new window for Algorithm 1 in [15]. We use Open image in new window as the initial point.

Example 4.2.

Then the set Open image in new window and the mapping Open image in new window satisfy the assumptions of Theorem 3.1 and (1,0,0,0) is a solution of the generalized variational inequality. We choose Open image in new window , and Open image in new window for the two algorithms.

## Notes

### 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).

### References

- 1.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/s00186-005-0052-2MathSciNetCrossRefMATHGoogle Scholar - 2.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/S1052623499352656MathSciNetCrossRefMATHGoogle Scholar - 3.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_6CrossRefGoogle Scholar - 4.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 - 5.Fang SC, Peterson EL: Generalized variational inequalities.
*Journal of Optimization Theory and Applications*1982, 38(3):363–383. 10.1007/BF00935344MathSciNetCrossRefMATHGoogle Scholar - 6.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/BF02592328MathSciNetCrossRefMATHGoogle Scholar - 7.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.2006.07.063MathSciNetCrossRefMATHGoogle Scholar - 8.Saigal R: Extension of the generalized complementarity problem.
*Mathematics of Operations Research*1976, 1(3):260–266. 10.1287/moor.1.3.260MathSciNetCrossRefMATHGoogle Scholar - 9.Salmon G, Strodiot J-J, Nguyen VH: A bundle method for solving variational inequalities.
*SIAM Journal on Optimization*2003, 14(3):869–893.MathSciNetCrossRefMATHGoogle Scholar - 10.Rockafellar RT: Monotone operators and the proximal point algorithm.
*SIAM Journal on Control and Optimization*1976, 14(5):877–898. 10.1137/0314056MathSciNetCrossRefMATHGoogle Scholar - 11.Konnov IV: On the rate of convergence of combined relaxation methods.
*Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika*1993, (12):89–92.Google Scholar - 12.Konnov IV:
*Combined Relaxation Methods for Variational Inequalities, Lecture Notes in Economics and Mathematical Systems*.*Volume 495*. Springer, Berlin, Germany; 2001:xii+181.CrossRefGoogle Scholar - 13.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.CrossRefGoogle Scholar - 14.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.2005.01.031MathSciNetCrossRefMATHGoogle Scholar - 15.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.2008.09.014MathSciNetCrossRefMATHGoogle Scholar - 16.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/S0363012997317475MathSciNetCrossRefMATHGoogle Scholar - 17.Facchinei F, Pang JS:
*Finite-Dimensional Variational Inequalities and Complementary Problems*. Springer, New York, NY, USA; 2003.MATHGoogle Scholar - 18.Karamardian S: Complementarity problems over cones with monotone and pseudomonotone maps.
*Journal of Optimization Theory and Applications*1976, 18(4):445–454. 10.1007/BF00932654MathSciNetCrossRefMATHGoogle Scholar - 19.Aubin J-P, Ekeland I:
*Applied Nonlinear Analysis, Pure and Applied Mathematics*. John Wiley & Sons, New York, NY, USA; 1984:xi+518.MATHGoogle Scholar

## Copyright information

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.