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A New Method for Solving Monotone Generalized Variational Inequalities

Abstract

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.

1. Introduction

Let be a convex subset of the real Euclidean space , be a continuous mapping from into , and be a lower semicontinuous convex function from into . We say that a point is a solution of the following generalized variational inequality if it satisfies

(GVI)

where denotes the standard dot product in .

Associated with the problem (GVI), the dual form of this is expressed as following which is to find such that

(DGVI)

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 [19]).

It is well known that the interior quadratic and dual technique are powerfull tools for analyzing and solving the optimization problems (see [1016]). Recently these techniques have been used to develop proximal iterative algorithm for variational inequalities (see [1722]).

In addition Nesterov [23] 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 [23] 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.

2. Preliminaries

First, let us recall the well-known concepts of monotonicity that will be used in the sequel (see [24]).

Definition 2.1.

Let be a convex set in , and . The function is said to be

(i)pseudomonotone on if

(2.1)

(ii)monotone on if for each ,

(2.2)

(iii)strongly monotone on with constant if for each ,

(2.3)

(iv)Lipschitz with constant on (shortly -Lipschitz), if

(2.4)

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]):

Find such that

(2.5)

Throughout this paper, we assume that:

(A 1 ) the interior set of , int  is nonempty,

(A 2 ) the set is bounded,

(A 3 ) is upper semicontinuous on , and is proper, closed convex and subdifferentiable on ,

(A 4 ) is monotone on .

In special case , problem (GVI) can be written by the following.

Find such that

(VI)

It is well known that the problem (VI) can be formulated as finding the zero points of the operator , where

(2.6)

The dual gap function of problem (GVI) is defined as follows:

(2.7)

The following lemma gives two basic properties of the dual gap function (2.7) whose proof can be found, for instance, in [6].

Lemma 2.2.

The function is a gap function of (GVI), that is,

(i) for all ,

(ii) and if and only if is a solution to (DGVI). Moreover, if is pseudomonotone then is a solution to (DGVI) if and only if it is a solution to (GVI).

The problem may not be solvable and the dual gap function may not be well-defined. Instead of using gap function , we consider a truncated dual gap function . Suppose that fixed and . The truncated dual gap function is defined as follows:

(2.8)

For the following consideration, we define as a closed ball in centered at and radius , and . The following lemma gives some properties for .

Lemma 2.3.

Under assumptions (A1)–(A4), the following properties hold.

(i)The function is well-defined and convex on .

(ii)If a point is a solution to (DGVI) then .

(iii)If there exists such that and , and is pseudomonotone, then is a solution to (DGVI) (and also (GVI)).

Proof.

  1. (i)

    Note that is upper semicontinuous on for and is bounded. Therefore, the supremum exists which means that is well-defined. Moreover, since is convex on and is the supremum of a parametric family of convex functions (which depends on the parameter ), then is convex on

  2. (ii)

    By definition, it is easy to see that for all . Let be a solution of (DGVI) and . Then we have

    (2.9)

In particular, we have

(2.10)

for all . Thus

(2.11)

this implies .

  1. (iii)

    For some , means that is a solution to (DGVI) restricted to . Since is pseudomonotone, is also a solution to (GVI) restricted to . Since , for any , we can choose sufficiently small such that

    (2.12)
(2.13)

where (2.13) follows from the convexity of . Since , dividing this inequality by , we obtain that is a solution to (GVI) on . Since is pseudomonotone, is also a solution to (DGVI).

Let be a nonempty, closed convex set and . Let us denote the Euclidean distance from to and the point attained this distance, that is,

(2.14)

As usual, is referred to the Euclidean projection onto the convex set . It is well-known that is a nonexpansive and co-coercive operator on (see [27, 28]).

The following lemma gives a tool for the next discussion.

Lemma 2.4.

For any and for any , the function and the mapping defined by (2.14) satisfy

(2.15)
(2.16)
(2.17)

Proof.

Inequality (2.15) is obvious from the property of the projection (see [27]). Now, we prove the inequality (2.16). For any , applying (2.15) we have

(2.18)

Using the definition of and noting that and taking minimum with respect to in (2.18), then we have

(2.19)

which proves (2.16).

From the definition of , we have

(2.20)

Since , applying (2.15) with instead of and for (2.20), we obtain the last inequality in Lemma 2.4.

For a given integer number , we consider a finite sequence of arbitrary points , a finite sequence of arbitrary points and a finite positive sequence . Let us define

(2.21)

Then upper bound of the dual gap function is estimated in the following lemma.

Lemma 2.5.

Suppose that Assumptions (A1)–(A4) are satisfied and

(2.22)

Then, for any ,

(i), for all , .

(ii).

Proof.

  1. (i)

    We define as the Lagrange function of the maximizing problem . Using duality theory in convex optimization, then we have

    (2.23)

 (ii) From the monotonicity of and (2.22), we have

(2.24)

Combining (2.24), Lemma 2.5(i) and

(2.25)

we get

(2.26)

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.

The dual interior proximal step at the iteration is generated by using the following scheme:

(3.1)

where and are given parameters, is the solution to (2.22).

The following lemma shows an important property of the sequence .

Lemma 3.1.

The sequence generated by scheme (3.1) satisfies

(3.2)

where , and . As a consequence, we have

(3.3)

Proof.

We replace by and by into (2.16) to obtain

(3.4)

Using the inequality (3.4) with , , and noting that , we get

(3.5)

This implies that

(3.6)

From the subdifferentiability of the convex function to scheme (3.1), using the first-order necessary optimality condition, we have

(3.7)

for all . This inequality implies that

(3.8)

where .

We apply inequality (3.4) with , and and using (3.8) to obtain

(3.9)

Combine this inequality and (3.6), we get

(3.10)

On the other hand, if we denote , then it follows that

(3.11)

Combine (3.10) and (3.11), we get

(3.12)

which proves (3.2).

On the other hand, from (3.9) we have

(3.13)

Then the inequality (3.3) is deduced from this inequality and (3.6).

The dual algorithm is an iterative method which generates a sequence based on scheme (3.1). The algorithm is presented in detail as follows:

Algorithm 3.2.

One has the following.

Initialization:

Given a tolerance , fix an arbitrary point and choose , . Take and .

Iterations:

For each , execute four steps below.

Step 1.

Compute a projection point by taking

(3.14)

Step 2.

Solve the strongly convex programming problem

(3.15)

to get the unique solution .

Step 3.

Find such that

(3.16)

Set .

Step 4.

Compute

(3.17)

If , where is a given tolerance, then stop.

Otherwise, increase by 1 and go back to Step 1.

Output:

Compute the final output as:

(3.18)

Now, we prove the convergence of Algorithm 3.2 and estimate its complexity.

Theorem 3.3.

Suppose that assumptions (A1)–(A3) are satisfied and is -Lipschitz continuous on . Then, one has

(3.19)

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 .

Proof.

From , where and , we get

(3.20)

Substituting (3.20) into (3.2), we obtain

(3.21)

Using this inequality with for all and , we obtain

(3.22)

If we choose for all in (2.21), then we have

(3.23)

Hence, from Lemma 2.5(ii), we have

(3.24)

Using inequality (3.22) and , it implies that

(3.25)

Note that . It follows from the inequalities (3.24) and (3.25) that

(3.26)

which implies that . The termination criterion at Step 4, , using inequality (2.26) we obtain and the number of iterations to reach an -solution is .

If there is no the guarantee for the Lipschitz condition, but the sequences and are uniformly bounded, we suppose that

(3.27)

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.

Algorithm 3.4.

One has the following.

Initialization:

Fix an arbitrary point and set . Take and . Choose for all .

Iterations:

For each execute the following steps.

Step 1.

Compute the projection point by taking

(3.28)

Step 2.

Solve the strong convex programming problem

(3.29)

to get the unique solution .

Step 3.

Find such that

(3.30)

Set .

Step 4.

Compute

(3.31)

If , where is a given tolerance, then stop.

Otherwise, increase by 1, update and go back to Step 1.

Output:

Compute the final output as

(3.32)

The next theorem shows the convergence of Algorithm 3.4.

Theorem 3.5.

Let assumptions (A1)–(A3) be satisfied and the sequence be generated by Algorithm 3.4. Suppose that the sequences and are uniformly bounded by (3.27). Then, we have

(3.33)

As a consequence, the sequence converges to 0 and the number of iterations to reach an -solution is .

Proof.

If we choose for all in (2.21), then we have . Since , it follows from Step 3 of Algorithm 3.4 that

(3.34)

From (3.34) and Lemma 2.5(ii), for all we have

(3.35)

We define . Then, we have

(3.36)

We consider, for all

(3.37)

Then derivative of is given by

(3.38)

Thus is nonincreasing. Combining this with (3.36) and , we have

(3.39)

From Lemma 3.1, and , we have

(3.40)

Combining (3.39) and this inequality, we have

(3.41)

By induction on , it follows from (3.41) and that

(3.42)

From (3.35) and (3.42), we obtain

(3.43)

which implies that . The remainder of the theorem is trivially follows from (3.33).

4. Illustrative Example and Numerical Results

In this section, we illustrate the proposed algorithms on a class of generalized variational inequalities (GVI), where is a polyhedral convex set given by

(4.1)

where , . The cost function is defined by

(4.2)

where , is a symmetric positive semidefinite matrix and . The function is defined by

(4.3)

Then is subdifferentiable, but it is not differentiable on .

For this class of problem (GVI) we have the following results.

Lemma 4.1.

Let . Then

(i)if is -strongly monotone on , then is monotone on whenever .

(ii)if is -strongly monotone on , then is -strongly monotone on whenever .

(iii)if is -Lipschitz on , then is -Lipschitz on .

Proof.

Since is -strongly monotone on , that is

(4.4)

we have

(4.5)

Then (i) and (ii) easily follow.

Using the Lipschitz condition, it is not difficult to obtain (iii).

To illustrate our algorithms, we consider the following data.

(4.6)

with , , . From Lemma 4.1, we have is monotone on . The subproblems in Algorithm 3.2 can be solved efficiently, for example, by using MATLAB Optimization Toolbox R2008a. We obtain the approximate solution

(4.7)

Now we use Algorithm 3.4 on the same variational inequalities except that

(4.8)

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 [19]. Under these assumptions, it can be proved that is continuous and monotone on .

With and the tolerance , we obtained the computational results (see, the Table 1).

Table 1 Numerical results: Algorithm 3.4 with .

References

  1. Anh PN, Muu LD, Strodiot J-J: Generalized projection method for non-Lipschitz multivalued monotone variational inequalities. Acta Mathematica Vietnamica 2009, 34(1):67–79.

    MathSciNet  MATH  Google Scholar 

  2. Anh PN, Muu LD, Nguyen VH, Strodiot JJ: Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities. Journal of Optimization Theory and Applications 2005, 124(2):285–306. 10.1007/s10957-004-0926-0

    MathSciNet  Article  MATH  Google Scholar 

  3. Bello Cruz JY, Iusem AN: Convergence of direct methods for paramontone variational inequalities. Computational Optimization and Applications 2010, 46(2):247–263. 10.1007/s10589-009-9246-5

    MathSciNet  Article  MATH  Google Scholar 

  4. Facchinei F, Pang JS: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, New York, NY, USA; 2003.

    MATH  Google Scholar 

  5. Fukushima M: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming 1992, 53(1):99–110. 10.1007/BF01585696

    MathSciNet  Article  MATH  Google Scholar 

  6. Konnov IV: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin, Germany; 2000.

    MATH  Google Scholar 

  7. Mashreghi J, Nasri M: Forcing strong convergence of Korpelevich's method in Banach spaces with its applications in game theory. Nonlinear Analysis: Theory, Methods & Applications 2010, 72(3–4):2086–2099. 10.1016/j.na.2009.10.009

    MathSciNet  Article  MATH  Google Scholar 

  8. Noor MA: Iterative schemes for quasimonotone mixed variational inequalities. Optimization 2001, 50(1–2):29–44. 10.1080/02331930108844552

    MathSciNet  Article  MATH  Google Scholar 

  9. Zhu DL, Marcotte P: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM Journal on Optimization 1996, 6(3):714–726. 10.1137/S1052623494250415

    MathSciNet  Article  MATH  Google Scholar 

  10. Daniele P, Giannessi F, Maugeri A: Equilibrium Problems and Variational Models, Nonconvex Optimization and Its Applications. Volume 68. Kluwer Academic Publishers, Norwell, Mass, USA; 2003:xiv+445.

    Book  MATH  Google Scholar 

  11. Fang SC, Peterson EL: Generalized variational inequalities. Journal of Optimization Theory and Applications 1982, 38(3):363–383. 10.1007/BF00935344

    MathSciNet  Article  MATH  Google Scholar 

  12. Goh CJ, Yang XQ: Duality in Optimization and Variational Inequalities, Optimization Theory and Applications. Volume 2. Taylor & Francis, London, UK; 2002:xvi+313.

    Book  MATH  Google Scholar 

  13. Iusem AN, Nasri M: Inexact proximal point methods for equilibrium problems in Banach spaces. Numerical Functional Analysis and Optimization 2007, 28(11–12):1279–1308. 10.1080/01630560701766668

    MathSciNet  Article  MATH  Google Scholar 

  14. Kim JK, Kim KS: New systems of generalized mixed variational inequalities with nonlinear mappings in Hilbert spaces. Journal of Computational Analysis and Applications 2010, 12(3):601–612.

    MathSciNet  MATH  Google Scholar 

  15. Kim JK, Kim KS: A new system of generalized nonlinear mixed quasivariational inequalities and iterative algorithms in Hilbert spaces. Journal of the Korean Mathematical Society 2007, 44(4):823–834. 10.4134/JKMS.2007.44.4.823

    MathSciNet  Article  MATH  Google Scholar 

  16. Waltz RA, Morales JL, Nocedal J, Orban D: An interior algorithm for nonlinear optimization that combines line search and trust region steps. Mathematical Programming 2006, 107(3):391–408. 10.1007/s10107-004-0560-5

    MathSciNet  Article  MATH  Google Scholar 

  17. Anh PN: An interior proximal method for solving monotone generalized variational inequalities. East-West Journal of Mathematics 2008, 10(1):81–100.

    MathSciNet  MATH  Google Scholar 

  18. Auslender A, Teboulle M: Interior projection-like methods for monotone variational inequalities. Mathematical Programming 2005, 104(1):39–68. 10.1007/s10107-004-0568-x

    MathSciNet  Article  MATH  Google Scholar 

  19. Bnouhachem A: An LQP method for pseudomonotone variational inequalities. Journal of Global Optimization 2006, 36(3):351–363. 10.1007/s10898-006-9013-4

    MathSciNet  Article  MATH  Google Scholar 

  20. Iusem AN, Nasri M: Augmented Lagrangian methods for variational inequality problems. RAIRO Operations Research 2010, 44(1):5–25. 10.1051/ro/2010006

    MathSciNet  Article  MATH  Google Scholar 

  21. Kim JK, Cho SY, Qin X: Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings. Journal of Inequalities and Applications 2010, 2010:-17.

    Google Scholar 

  22. Kim JK, Buong N: Regularization inertial proximal point algorithm for monotone hemicontinuous mapping and inverse strongly monotone mappings in Hilbert spaces. Journal of Inequalities and Applications 2010, 2010:-10.

    Google Scholar 

  23. Nesterov Y: Dual extrapolation and its applications to solving variational inequalities and related problems. Mathematical Programming 2007, 109(2–3):319–344. 10.1007/s10107-006-0034-z

    MathSciNet  Article  MATH  Google Scholar 

  24. Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1984:xi+518.

    MATH  Google Scholar 

  25. Anh PN, Muu LD: Coupling the Banach contraction mapping principle and the proximal point algorithm for solving monotone variational inequalities. Acta Mathematica Vietnamica 2004, 29(2):119–133.

    MathSciNet  MATH  Google Scholar 

  26. Cohen G: Auxiliary problem principle extended to variational inequalities. Journal of Optimization Theory and Applications 1988, 59(2):325–333.

    MathSciNet  MATH  Google Scholar 

  27. Mangasarian OL, Solodov MV: A linearly convergent derivative-free descent method for strongly monotone complementarity problems. Computational Optimization and Applications 1999, 14(1):5–16. 10.1023/A:1008752626695

    MathSciNet  Article  MATH  Google Scholar 

  28. Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization 1976, 14(5):877–898. 10.1137/0314056

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

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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Anh, P., Kim, J. A New Method for Solving Monotone Generalized Variational Inequalities. J Inequal Appl 2010, 657192 (2010). https://doi.org/10.1155/2010/657192

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Keywords

  • Variational Inequality
  • Variational Inequality Problem
  • Projection Point
  • Polyhedral Convex
  • Convex Programming Problem