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Regularization and Iterative Methods for Monotone Variational Inequalities
Fixed Point Theory and Applications volume 2010, Article number: 765206 (2009)
We provide a general regularization method for monotone variational inequalities, where the regularizer is a Lipschitz continuous and strongly monotone operator. We also introduce an iterative method as discretization of the regularization method. We prove that both regularization and iterative methods converge in norm.
where is a nonempty closed convex subset of a real Hilbert space with inner product and norm , respectively, and is a monotone operator in with domain .
Recall that is monotone if
A typical example of monotone operators is the subdifferential of a proper convex lower semicontinuous function.
Variational inequality problems are equivalent to fixed point problems. As a matter of fact, solves VIP (1.1) if and only if solves the following fixed point problem (FPP), for any ,
where is the metric (or nearest point) projection from onto ; namely, for each , is the unique point in with the property
The equivalence between VIP (1.1) and FPP (1.3) is an immediate consequence of the following characterization of :
The dual VIP of (1.1) is the following VIP:
The following equivalence between the dual VIP (1.6) and the primal VIP (1.1) plays a useful role in our regularization in Section 2.
Lemma 1.1 (cf. ).
Assume that is monotone and weakly continuous along segments (i.e., weakly as for ), then the dual VIP (1.6) is equivalent to the primal VIP (1.1).
To guarantee the existence and uniqueness of a solution of VIP (1.1), one has to impose conditions on the operator . The following existence and uniqueness result is well known.
If is Lipschitz continuous and strongly monotone, then there exists one and only one solution to VIP (1.1).
However, if fails to be Lipschitz continuous or strongly monotone, then the result of the above theorem is false in general. We will assume that is Lipschitz continuous, but do not assume strong monotonicity of . Thus, VIP (1.1) is ill-posed and regularization is needed; moreover, a solution is often sought through iteration methods.
In the special case where is of the form , with being a nonexpansive mapping, regularization and iterative methods for VIP (1.1) have been investigated in literature; see, for example, [5–19]; work related to variational inequalities of monotone operators can be found in [20–25], and work related to iterative methods for nonexpansive mappings can be found in [26–33].
The aim of this paper is to provide a regularization and its induced iteration method for VIP (1.1) in the general case. The paper is structured as follows. In the next section we present a general regularization method for VI (1.1) with the regularizer being a Lipschitz continuous and strongly monotone operator. In Section 3, by discretizing the implicit method of the regularization obtained in Section 2, we introduce an iteration process and prove its strong convergence. In the final section, Section 4, we apply the results obtained in Sections 2 and 3 to a convex minimization problem.
Since VIP (1.1) is usually ill-posed, regularization is necessary, towards which we let be a Lipschitz continuous, everywhere defined, strongly monotone, and single-valued operator. Consider the following regularized variational inequality problem:
Since is strongly monotone, VI (2.1) has a unique solution which is denoted by . Indeed, VI (2.1) is equivalent to the fixed point equation
where , with .
To analyze more details of VI (2.1) (or its equivalent fixed point equation (2.2)), we need to impose more assumptions on the operators and . Assume that and are Lipschitz continuous with Lipschiz constants , respectively. We also assume that is -strongly monotone; namely, there is a constant satisfying the property
If is chosen in such a way that
then is a contraction with contraction coefficient
hence, is a -contraction.
Noticing that is -Lipschitzian and -strongly monotone, we deduce that, for ,
It turns out that if satisfies (2.4), then is a contraction with coefficient given by the left side of (2.5).
Finally, it is straightforward that (2.7) holds provided that satisfies (2.6).
Below we always assume that satisfies (2.6) so that is a -contraction from into itself. Therefore, for such a choice of , has a unique fixed point in which is denoted as whose asymptotic behavior when is given in the following result.
(a) is monotone on and weakly continuous along segments in (i.e., weakly as for ),
(b) is -monotone on ,
(c)the solution set of VI (1.1) is nonempty.
For , let be the unique solution of the regularized VIP (2.1). Then, as , converges in norm to a point in which is the unique solution of the VIP
Therefore, if one takes to be the identity operator, then the regularized solution of the corresponding regularized VIP (2.1) converges in norm to the minimal norm point of the solution set .
To prove Theorem 2.2, we first prove the boundedness of the net .
Assume that is monotone on . Assume conditions (b) and (c) in Theorem 2.2. Then is bounded; indeed, for any ,
We have (2.1) holds for all . In particular, for , we have
It turns out that
Since is monotone and is -strongly monotone, we have
Substituting them into (2.12) we obtain
However, since , . We therefore get from (2.14) that
Now (2.10) follows immediately from (2.15).
Proof of Theorem 2.2.
Since is bounded by Lemma 2.3, the set of weak limit points as of the net , , is nonempty. Pick a and let be a null sequence in the interval such that weakly as . We first show that . To see this we use the equivalent dual VI of (2.1):
Thus, we have, for all and ,
Taking the limit as yields that
It turns out that .
We next prove that the sequence actually converges to strongly. Replacing in (2.15) with gives
Now it is straightforward from (2.19) that the weak convergence to of implies strong convergence to of .
The relation (2.15) particularly implies that, for ,
which in turns implies that every point solves the VIP
or equivalently, the VIP
However, since is strongly monotone, the solution to VIP (2.22) is unique. This has shown that the unique solution of VIP (2.22) is the strong limit of the net .
Finally, if is the identity operator, then VIP (2.22) is reduced to
This is equivalent to
which immediately implies that for all and hence is the minimal norm of .
In Theorem 2.2, we have proved that if the solution set of VIP (1.1) is nonempty, then the net of the solutions of the regularized VIPs (2.1) is bounded (and hence converges in norm). The converse is indeed also true; that is, the boundedness of the net implies that the solution set of VIP (1.1) is nonempty. As a matter of fact, suppose that is bounded and is a constant such that for all .
By Lemma 1.1, we have
Since is bounded, we can easily see that every weak cluster point of the net solves the VIP
This is the dual VI to the primal VI (2.1); hence is a solution of VI (2.1) by Lemma 1.1.
3. Iterative Method
From the fixed point equation (2.2), it is natural to consider the following iteration method that generates a sequence according to the recursion:
where the initial guess is selected arbitrarily, and and are two sequences of positive numbers in . Put in another way, is the unique solution in of the following VIP:
(a) is -Lipschitz continuous and monotone on ,
(b) is -Lipschitz continuous and -monotone on
(c)the solution set of VI (1.1) is nonempty.
Assume in addition that
(ii) as ,
then the sequence generated by the algorithm (3.1) converges in norm to the unique solution of VI (2.9).
To prove Theorem 3.1, we need a lemma below.
Lemma 3.2 (cf. ).
Assume that is a sequence of nonnegative real numbers such that
where and are real sequences such that
(i) for all , and ;
Proof of Theorem 3.1.
Let , where . By assumption (i) and Lemma 2.1, is a contraction and has a unique fixed point which is denoted by . Moreover, by Theorem 2.2, converges in norm to the unique solution of VI (2.9). Therefore, it suffices to prove that as .
To see this, observing that is a -contraction, we obtain
However, we have
Since is bounded, it turns out that, for an appropriate constant ,
Substituting (3.6) into (3.4) and setting , we get
with . Assumptions (iii) and (iv) assure that and as , respectively. Therefore, we can apply lemma to (3.7) to conclude that ; hence, in norm.
Assume satisfy , then it is not hard to see that for an appropriate constant ,
satisfy the assumptions (i)–(iv) of Theorem 3.1.
Consider the constrained convex minimization problem:
where is a closed convex subset of a real Hilbert space and is a real-valued convex function. Assume that is continuously differentiable with a Lipschitz continuous gradient:
where is a constant.
It is known that the minimization (4.1) is equivalent to the variational inequality problem:
Therefore, applying Theorems 2.2 and 3.1, we get the following result.
Assume the Lipschitz continuity (4.2) for the gradient .
For , let be the unique solution of the regularized VIP
Equivalently, is the unique solution in of the regularized minimization problem:
Then, as , remains bounded if and only if (4.1) has a solution, and in this case, converges in norm to the minimal norm solution of (4.1).
Assume that (4.1) has a solution. Assume in addition that
(ii) as ,
Starting , one defines by the iterative algorithm
Then converges in norm to the minimum-norm solution of the constrained minimization problem (4.1).
Apply Theorems 2.2 and 3.1 to the case where and is the identity operator to get the conclusions in (a) and (b).
Baiocchi C, Capelo A: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1984:ix+452.
Giannessi F, Maugeri A: Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York, NY, USA; 1995.
Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics. Volume 88. Academic Press, New York, NY, USA; 1980:xiv+313.
Konnov IV, Schaible S: Duality for equilibrium problems under generalized monotonicity. Journal of Optimization Theory and Applications 2000,104(2):395–408. 10.1023/A:1004665830923
Ceng L-C, Ansari QH, Yao J-C: Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces. Numerical Functional Analysis and Optimization 2008,29(9–10):987–1033. 10.1080/01630560802418391
Zeng LC, Schaible S, Yao J-C: Hybrid steepest descent methods for zeros of nonlinear operators with applications to variational inequalities. Journal of Optimization Theory and Applications 2009,141(1):75–91. 10.1007/s10957-008-9501-4
Ceng L-C, Xu HK, Yao J-C: A hybrid steepest-descent method for variational inequalities in Hilbert spaces. Applicable Analysis 2008,87(5):575–589. 10.1080/00036810802140608
Chen R, Su Y, Xu HK: Regularization and iteration methods for a class of monotone variational inequalities. Taiwanese Journal of Mathematics 2009,13(2B):739–752.
Lemaire B, Salem ChOA: From fixed-point regularization to constraint decomposition in variational inequalities. In Proceedings of the 9th Belgian-French-German Conference on Optimization, September 1998, Namur, Belgium, Lecture Notes in Economic and Mathematical Systems. Volume 481. Springer; 250–259.
Lu X, Xu H-K, Yin X: Hybrid methods for a class of monotone variational inequalities. Nonlinear Analysis: Theory, Methods & Applications 2009,71(3–4):1032–1041. 10.1016/j.na.2008.11.067
Maingé P-E, Moudafi A: Strong convergence of an iterative method for hierarchical fixed-point problems. Pacific Journal of Optimization 2007,3(3):529–538.
Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006,318(1):43–52. 10.1016/j.jmaa.2005.05.028
Marino G, Xu HK: Explicit hierarchical fixed point approach to variational inequalities. preprint
Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615
Moudafi A, Maingé P-E: Towards viscosity approximations of hierarchical fixed-point problems. Fixed Point Theory and Applications 2006, 2006:-10.
Salem ChOA: Approximation de points fixes d'une contraction, Ph.D. thesis. Université Montpellier, Cedex, France; 1998.
Tam NN, Yao J-C, Yen ND: Solution methods for pseudomonotone variational inequalities. Journal of Optimization Theory and Applications 2008,138(2):253–273. 10.1007/s10957-008-9376-4
Wong NC, Sahu DR, Yao J-C: Solving variational inequalities involving nonexpansive type mappings. Nonlinear Analysis: Theory, Methods & Applications 2008,69(12):4732–4753. 10.1016/j.na.2007.11.025
Yao J-C: Variational inequalities with generalized monotone operators. Mathematics of Operations Research 1994,19(3):691–705. 10.1287/moor.19.3.691
Xu HK: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002,66(1):240–256. 10.1112/S0024610702003332
Xu HK: Another control condition in an iterative method for nonexpansive mappings. Bulletin of the Australian Mathematical Society 2002,65(1):109–113. 10.1017/S0004972700020116
Xu HK: Remarks on an iterative method for nonexpansive mappings. Communications on Applied Nonlinear Analysis 2003,10(1):67–75.
Xu HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003,116(3):659–678. 10.1023/A:1023073621589
Xu HK: Viscosity method for hierarchical fixed point approach to variational inequalities. Taiwanese Journal of Mathematics 2010.,14(2):
Yamada I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Stud. Comput. Math.. Volume 8. North-Holland, Amsterdam, The Netherlands; 2001:473–504.
Browder FE: Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces. Archive for Rational Mechanics and Analysis 1967, 24: 82–90.
Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0
Lions P-L: Approximation de points fixes de contractions. Comptes Rendus de l'Académie des Sciences de Paris, Série. A-B 1977,284(21):A1357-A1359.
Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. Journal of Mathematical Analysis and Applications 1980,75(1):287–292. 10.1016/0022-247X(80)90323-6
Reich S: Approximating fixed points of nonexpansive mappings. PanAmerican Mathematical Journal 1994,4(2):23–28.
Shioji N, Takahashi W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proceedings of the American Mathematical Society 1997,125(12):3641–3645. 10.1090/S0002-9939-97-04033-1
Wittmann R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992,58(5):486–491. 10.1007/BF01190119
Xu HK: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059
The authors are grateful to the anonymous referees for their comments and suggestions which improved the presentation of this paper. This paper is dedicated to Professor William Art Kirk for his significant contributions to fixed point theory. The first author was supported in part by a fund (Grant no. 2008ZG052) from Zhejiang Administration of Foreign Experts Affairs. The second author was supported in part by NSC 97-2628-M-110-003-MY3, and by DGES MTM2006-13997-C02-01.
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Xu, X., Xu, H. Regularization and Iterative Methods for Monotone Variational Inequalities. Fixed Point Theory Appl 2010, 765206 (2009). https://doi.org/10.1155/2010/765206
- Variational Inequality
- Iterative Method
- Nonexpansive Mapping
- Monotone Operator
- Variational Inequality Problem