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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 458134, 20 pages doi:10.1155/2009/458134 Research Article Self-Adaptive Implicit Methods for Monotone Variant Variational Inequalities Zhili Ge and Deren Han Institute of Mathematics, School of Mathematics and Computer Science, Nanjing Normal University, Nanjing 210097, China Correspondence should be addressed to Deren Han, handr00@hotmail.com Received 26 January 2009; Accepted 24 February 2009 Recommended by Ram U. Verma The efficiency of the implicit method proposed by He 1999 dependsontheparameterβ heavily; while it varies for individual problem, that is, different problem has different “suitable” parameter, whichisdifficult to find. In this paper, we present a modified implicit method, which adjusts the parameter β automatically per iteration, based on the message from former iterates. To improve the performance of the algorithm, an inexact version is proposed, where the subproblem is just solved approximately. Under mild conditions as those for variational inequalities, we prove the global convergence of both exact and inexact versions of the new method. We also present several preliminary numerical results, which demonstrate that the self-adaptive implicit method, especially the inexact version, is efficient and robust. Copyright q 2009 Z. Ge and D. Han. 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. 1. Introduction Let Ω be a closed convex subset of R n and let F be a mapping from R n into itself. The so-called finite-dimensional variant variational inequalities, denoted by VVIΩ,F, is to find a vector u ∈R n , such that F  u  ∈ Ω,  v − F  u   u ≥ 0, ∀v ∈ Ω, 1.1 while a classical variational inequality problem, abbreviated by VIΩ,f,istofindavector x ∈ Ω, such that  x  − x   f  x  ≥ 0, ∀x  ∈ Ω, 1.2 where f is a mapping from R n into itself. 2 Journal of Inequalities and Applications Both VVIΩ,F and VIΩ,f serve as very general mathematical models of numerous applications arising in economics, engineering, transportation, and so forth. They include some widely applicable problems as special cases, such as mathematical programming problems, system of nonlinear equations, and nonlinear complementarity problems, and so forth. Thus, they have been extensively investigated. We refer the readers to the excellent monograph of Faccinei and Pang 1, 2 and the references therein for theoretical and algorithmic developments on VIΩ,f, for example, 3–10,and11–16 for VVIΩ,F. It is observed that if F is invertible, then by setting f  F −1 , the inverse mapping of F, VVIΩ,F can be reduced to VIΩ,f. Thus, theoretically, all numerical methods for solving VIΩ,f can be used to solve VVIΩ,F. However, in many practical applications, the inverse mapping F −1 may not exist. On the other hand, even if it exists, it is not easy to find it. Thus, there is a need to develop numerical methods for VVIΩ,F and recently, the Goldstein’s type method was extended from solving VIΩ,f to VVIΩ,F12, 17. In 11, He proposed an implicit method for solving general variational inequality problems. A general variational inequality problem is to find a vector u ∈R n , such that F  u  ∈ Ω,  v − Fu   G  u  ≥ 0, ∀v ∈ Ω. 1.3 When G is the identity mapping, it reduces to VVIΩ,F and if F is the identity mapping, it reduces to VIΩ,G. He’s implicit method is as follows. S0 Given u 0 ∈ R n ,β>0,γ∈ 0, 2, and a positive definite matrix M. S1 Find u k1 via θ k  u   0, 1.4 where θ k  u   F  u   βG  u  − F  u k  − βG  u k   γρ  u k ,M,β  M −1 e  u k ,β  , 1.5 ρ  u k ,M,β     eu k ,β   2 eu k ,β  M −1 e  u k ,β  , e  u, β  : F  u  − P Ω  F  u  − βG  u   , 1.6 with P Ω being the projection from R n onto Ω, under the Euclidean norm. He’s method is attractive since it solves the general variational inequality problem, which is essentially equivalent to a system of nonsmooth equations e  u, β   0, 1.7 via solving a series of smooth equations 1.4. The mapping in the subproblem is well conditioned and many efficient numerical methods, such as Newton’s method, can be applied Journal of Inequalities and Applications 3 to solve it. Furthermore, to improve the efficiency of the algorithm, He 11 proposed to solve the subproblem approximately. That is, at Step 1, instead of finding a zero of θ k , it only needs to find a vector u k1 satisfying    θ k  u k1     ≤ η k    e  u k ,β     , 1.8 where {η k } is a nonnegative sequence. He proved the global convergence of the algorithm under the condition that the error tolerance sequence {η k } satisfies ∞  k0 η k 2 < ∞. 1.9 In the above algorithm, there are two parameters β>0andγ ∈ 0, 2, which affect the efficiency of the algorithm. It was observed that nearly for all problems, γ close to 2 is a better choice than smaller γ, while different problem has different optimalβ. A suitable parameter β is thus difficult to find for an individual problem. For solving variational inequality problems, He et al. 18 proposed to choose a sequence of parameters {β k }, instead of a fixed parameter β, to improve the efficiency of the algorithm. Under the same conditions as those in 11, they proved the global convergence of the algorithm. The numerical results reported there indicated that for any given initial parameter β 0 , the algorithm can find a suitable parameter self-adaptively. This improves the efficiency of the algorithm greatly and makes the algorithm easy and robust to implement in practice. In this paper, in a similar theme as 18, we suggest a general rule f or choosing suitable parameter in the implicit method for solving VVIΩ,F. By replacing the constant f actor β in 1.4 and 1.5 with a self-adaptive variable positive sequence {β k },theefficiency of the algorithm can be improved greatly. Moreover, it is also robust to the initial choice of the parameter β 0 . Thus, for any given problems, we can choose a parameter β 0 arbitrarily, for example, β 0  1orβ 0  0.1. The algorithm chooses a suitable parameter self-adaptively, based on the information from the former iteration, which makes it able to add a little additional computational cost against the original algorithm with fixed parameter β.To further improve the efficiency of the algorithm, we also admit approximate computation in solving the subproblem per iteration. That is, per iteration, we just need to find a vector u k1 that satisfies 1.8. Throughout this paper, we make the following assumptions. Assumption A. The solution set of VVIΩ,F, denoted by Ω ∗ , is nonempty. Assumption B. The operator F is monotone, that is, for any u, v ∈R n , u − v   F  u  − F  v  ≥ 0. 1.10 The rest of this paper is organized as follows. In Section 2, we summarize some basic properties which are useful in the convergence analysis of our method. In Sections 3 and 4, we describe the exact version and inexact version of the method and prove their global convergence, respectively. We report our preliminary computational results in Section 5 and give some final conclusions in the last section. 4 Journal of Inequalities and Applications 2. Preliminaries For a vector x ∈R n and a symmetric positive definite matrix M ∈R n×n , we denote x  √ x  x as the Euclidean-norm and x M as the matrix-induced norm, that is, x M : x  Mx 1/2 . Let Ω be a nonempty closed convex subset of R n ,andletP Ω · denote the projection mapping from R n onto Ω, under the matrix-induced norm. That is, P Ω  x  : arg min    x − y   M ,y∈ Ω  . 2.1 It is known 12, 19 that the variant variational inequality problem 1.1 is equivalent to the projection equation F  u   P Ω  F  u  − βM −1 u  , 2.2 where β is an arbitrary positive constant. Then, we have the following lemma. Lemma 2.1. u ∗ is a solution of VVIΩ,F if and only if eu, β0 for any fixed constant β>0, where e  u, β  : F  u  − P Ω  F  u  − βM −1 u  2.3 is the residual function of the projection equation 2.2. Proof. See 11, Theorem 1. The following lemma summarizes some basic properties of the projection operator, which will be used in the subsequent analysis. Lemma 2.2. Let Ω be a closed convex set in R n and let P Ω denote the projection operator onto Ω under the matrix-induced norm, then one has w − P Ω v  M  v − P Ω  v  ≤ 0, ∀v ∈R n , ∀w ∈ Ω, 2.4  P Ω u − P Ω v  M ≤  u − v  M , ∀u, v ∈R n . 2.5 The following lemma plays an important role in convergence analysis of our algorithm. Lemma 2.3. For a given u ∈R n ,let  β ≥ β>0. Then it holds that    e  u,  β     M ≥   eu, β   M . 2.6 Proof. See 20 for a simple proof. Journal of Inequalities and Applications 5 Lemma 2.4. Let u ∗ ∈ Ω ∗ , then for all u ∈R n and β>0, one has {Fu − Fu ∗   βM −1 u − u ∗ }  Me  u, β  ≥   eu, β   2 M . 2.7 Proof. It follows from the definition of VVIΩ,Fsee 1.1 that {P Ω Fu − βM −1 u − Fu ∗ }  βu ∗ ≥ 0. 2.8 By setting v : Fu − βM −1 u and w : Fu ∗  in 2.4,weobtain {P Ω Fu − βM −1 u − Fu ∗ }  M  e  u, β  − βM −1 u  ≥ 0. 2.9 Adding 2.8 and 2.9, and using the definition of eu, β in 2.3,weget {Fu − Fu ∗  − eu, β}  M  e  u, β  − βM −1  u − u ∗   ≥ 0, 2.10 that is, Fu − Fu ∗ βM −1 u − u ∗   Me  u, β  ≥   eu, β   2 M  βFu − Fu ∗    u − u ∗  ≥   eu, β   2 M , 2.11 where the last inequality follows from the monotonicity of F Assumption B. This completes the proof. 3. Exact Implicit Method and Convergence Analysis We are now in the position to describe our algorithm formally. 3.1. Self-Adaptive Exact Implicit Method S0 Given γ ∈ 0, 2, β 0 > 0, u 0 ∈R n and a positive definite matrix M. S1 Compute u k1 such that F  u k1   β k M −1 u k1 − F  u k  − β k M −1 u k  γe  u k ,β k   0. 3.1 6 Journal of Inequalities and Applications S2 If the given stopping criterion is satisfied, then stop; otherwise choose a new parameter β k1 ∈ 1/1  τ k β k , 1  τ k β k , where τ k satisfies ∞  k0 τ k < ∞,τ k ≥ 0. 3.2 Set k : k  1 and go to Step 1. From 3.1, we know that u k1 is the exact unique zero of θ k  u  : F  u   β k M −1 u − F  u k  − β k M −1 u k  γe  u k ,β k  . 3.3 We refer to the above method as the self-adaptive exact implicit method. Remark 3.1. According to the assumption τ k ≥ 0and  ∞ k0 τ k < ∞, we have  ∞ k0 1  τ k  < ∞. Denote S τ : ∞  k0  1  τ k  . 3.4 Hence, the sequence{β k }⊂1/S τ β 0 ,S τ β 0  is bounded. Then, let inf{β k } ∞ k0 : β L > 0and sup{β k } ∞ k0 : β U < ∞. Now, we analyze the convergence of the algorithm, beginning with the following lemma. Lemma 3.2. Let {u k } be the sequence generated by the proposed self-adaptive exact implicit method. Then for any u ∗ ∈ Ω ∗ and k>0, one has    Fu k1  − Fu ∗   β k M −1 u k1 − u ∗     2 M ≤    Fu k  − Fu ∗   β k M −1 u k − u ∗     2 M − γ  2 − γ     eu k ,β k     2 M . 3.5 Proof. Using 3.1,weget    Fu k1  − Fu ∗   β k M −1 u k1 − u ∗     2 M     Fu k  − Fu ∗   β k M −1 u k − u ∗  − γeu k ,β k     2 M ≤    Fu k  − Fu ∗   β k M −1 u k − u ∗     2 M − 2γ    eu k ,β k     2 M  γ 2    eu k ,β k     2 M     Fu k  − Fu ∗   β k M −1 u k − u ∗     2 M − γ  2 − γ     eu k ,β k     2 M , 3.6 where the inequality follows from 2.7. This completes the proof. Journal of Inequalities and Applications 7 Since 0 <β k1 ≤ 1  τ k β k and F is monotone, it follows that    Fu k1  − Fu ∗   β k1 M −1 u k1 − u ∗     2 M     Fu k1  − Fu ∗   β k M −1 u k1 − u ∗ β k1 − β k M −1 u k1 − u ∗     2 M     Fu k1  − Fu ∗   β k M −1 u k1 − u ∗     2 M β k1 − β k  2    u k1 − u ∗     2 M  2  β k1 − β k  u k1 − u ∗    F  u k1  − F  u ∗    β k M −1  u k1 − u ∗  ≤ 1  τ k  2    Fu k1  − Fu ∗   β k M −1 u k1 − u ∗     2 M , 3.7 where the inequality follows from the monotonicity of the mapping F. Combining 3.5 and 3.7, we have    Fu k1  − Fu ∗   β k1 M −1 u k1 − u ∗     2 M ≤ 1  τ k  2    Fu k  − Fu ∗   β k M −1 u k − u ∗     2 M − γ  2 − γ     eu k ,β k     2 M . 3.8 Now, we give the self-adaptive rule in choosing the parameter β k . For the sake of balance, we hope that    Fu k1  − Fu k     M ≈    β k M −1 u k1 − u k     M . 3.9 That is, for given constant τ>0, if    Fu k1  − Fu k     M >  1  τ     β k M −1 u k1 − u k     M , 3.10 we should increase β k in the next iteration; on the other hand, we should decrease β k when    Fu k1  − Fu k     M < 1  1  τ     β k M −1 u k1 − u k     M . 3.11 Let ω k    Fu k1  − Fu k    M   β k M −1 u k1 − u k    M . 3.12 8 Journal of Inequalities and Applications Then we give β k1 : ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩  1  τ k  β k , if ω k >  1  τ  , 1  1  τ k  β k , if ω k < 1  1  τ  , β k , otherwise. 3.13 Such a self-adaptive strategy was adopted in 18, 21–24 for solving variational inequality problems, where the numerical results indicated its efficiency and robustness to the choice of the initial parameter β 0 . Here we adopted it for solving variant variational inequality problems. We are now in the position to give the convergence result of the algorithm, the main result of this section. Theorem 3.3. The sequence {u k } generated by the proposed self-adaptive exact implicit method converges to a solution of VVIΩ,F. Proof. Let ξ k : 2τ k  τ k 2 . Then from the assumption that  ∞ k0 τ k < ∞, we have that  ∞ k0 ξ k < ∞, which means that  ∞ k0 1  ξ k  < ∞. Denote C s : ∞  i0 ξ i ,C p : ∞  i0  1  ξ i  . 3.14 From 3.8, for any u ∗ ∈ Ω ∗ , that is, an arbitrary solution of VVIΩ,F, we have    Fu k1  − Fu ∗   β k1 M −1 u k1 − u ∗     2 M ≤  1  ξ k     Fu k  − Fu ∗   β k M −1 u k − u ∗     2 M ≤  k  i0  1  ξ i      Fu 0  − Fu ∗   β 0 M −1 u 0 − u ∗     2 M 3.15 ≤ C p    Fu 0  − Fu ∗   β 0 M −1 u 0 − u ∗     2 M < ∞. 3.16 This, together with the monotonicity of the mapping F, means that the generated sequence {u k } is bounded. Journal of Inequalities and Applications 9 Also from 3.8, we have γ  2 − γ     eu k ,β k     2 M ≤ 1  τ k  2    Fu k  − Fu ∗   β k M −1 u k − u ∗     2 M −    Fu k1  − Fu ∗   β k1 M −1 u k1 − u ∗     2 M     Fu k  − Fu ∗   β k M −1 u k − u ∗     2 M −    Fu k1  − Fu ∗   β k1 M −1 u k1 − u ∗     2 M  ξ k    Fu k  − Fu ∗   β k M −1 u k − u ∗     2 M . 3.17 Adding both sides of the above inequality, we obtain γ  2 − γ  ∞  kk 0    eu k ,β k     2 M ≤    Fu 0  − Fu ∗   β 0 M −1 u 0 − u ∗     2 M  ∞  k0 ξ k    Fu k  − Fu ∗   β k M −1 u k − u ∗     2 M ≤    Fu 0  − Fu ∗   β 0 M −1 u 0 − u ∗     2 M   ∞  k0 ξ k  C p    Fu 0  − Fu ∗   β 0 M −1 u 0 − u ∗     2 M   1  C s C p     Fu 0  − Fu ∗   β 0 M −1 u 0 − u ∗     2 M < ∞, 3.18 where the second inequality follows from 3.15. Thus, we have lim k →∞    eu k ,β k     M  0, 3.19 which, from Lemma 2.3, means that lim k →∞    eu k ,β L     M ≤ lim k →∞    eu k ,β k     M  0. 3.20 10 Journal of Inequalities and Applications Since {u k } is bounded, it has at least one cluster point. Let u be a cluster point of {u k } and let {u k j } be the subsequence converging to u. Since eu, β L  is continuous, taking limit in 3.20 along the subsequence, we get   e u, β L    M  lim j →∞    eu k j ,β L     M  0. 3.21 Thus, from Lemma 2.1, u is a solution of VVIΩ,F. In the following we prove that the sequence {u k }has exactly one cluster point. Assume that u is another cluster point of {u k }, which is different from u. Because u is a cluster point of the sequence {u k } and F is monotone, there is a k 0 > 0 such that    Fu k 0  − Fuβ k 0 M −1 u k 0 − u    M ≤ δ 2C p , 3.22 where δ :    F  u  − F  u  β k 0 M −1 u − u    M . 3.23 On the other hand, since u ∈ Ω ∗ and u ∗ is an arbitrary solution, by setting u ∗ : u in 3.15, we have for all k ≥ k 0 ,    Fu k  − Fu  β k M −1 u k − u    2 M ≤ k  ik 0  1  ξ i     Fu i  − Fu  β i M −1 u i − u    2 M ≤ C p    Fu k 0  − Fu  β k 0 M −1 u k 0 − u    2 M , 3.24 that is,    Fu k  − Fu  β k M −1 u k − u    M ≤ C 1/2 p    Fu k 0  − Fu  β k 0 M −1 u k 0 − u    M ≤ δ 2C 1/2 p . 3.25 [...]... nonexpansive mappings and solutions of variational inequalities,” Journal of Inequalities and Applications, vol 2008, Article ID 284345, 12 pages, 2008 11 B S He, “Inexact implicit methods for monotone general variational inequalities,” Mathematical Programming, vol 86, no 1, pp 199–217, 1999 12 B S He, “A Goldstein’s type projection method for a class of variant variational inequalities,” Journal of... the choice of the initial point u0 6 Conclusions In this paper, we proposed a self-adaptive implicit method for solving monotone variant variational inequalities The proposed self-adaptive adjusting rule avoids the difficult task of choosing a “suitable” parameter, which makes the method efficient for initial parameter Our self-adaptive rule adds only a tiny amount of computation than the method with... vol 33, no 1, pp 168–184, 1995 17 M Li and X M Yuan, “An improved Goldstein’s type method for a class of variant variational inequalities,” Journal of Computational and Applied Mathematics, vol 214, no 1, pp 304–312, 2008 18 B S He, L Z Liao, and S L Wang, Self-adaptive operator splitting methods for monotone variational inequalities,” Numerische Mathematik, vol 94, no 4, pp 715–737, 2003 19 B C Eaves,... 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Therefore, we test the problem with α κ Ac and otherwise u κ ∈ 0, 1 In the test we take γ 1.85, τk 0.85, u0 0, and β0 0.1 The stopping criterion is e uk , βk α ≤ 10−8 5.14 The results in Table 1 show that β0 0.1 is a “proper” parameter for the problem with κ 0.05, while for the other two cases with larger κ 0.5 and with smaller κ 0.01, it is not For any of these three cases, the method with self-adaptive. .. Peˇ ari´ , “On some inequalities for beta and gamma functions via c c c some classical inequalities,” Journal of Inequalities and Applications, vol 2005, no 5, pp 593–613, 2005 6 S Dafermos, “Traffic equilibrium and variational inequalities,” Transportation Science, vol 14, no 1, pp 42–54, 1980 7 R U Verma, “A class of projection-contraction methods applied to monotone variational inequalities,” Applied... 2002 22 D Han and W Sun, “A new modified Goldstein-Levitin-Polyak projection method for variational inequality problems,” Computers & Mathematics with Applications, vol 47, no 12, pp 1817–1825, 2004 20 Journal of Inequalities and Applications 23 D Han, “Inexact operator splitting methods with selfadaptive strategy for variational inequality problems,” Journal of Optimization Theory and Applications,... D Han, W Xu, and H Yang, “An operator splitting method for variational inequalities with partially unknown mappings,” Numerische Mathematik, vol 111, no 2, pp 207–237, 2008 25 R S Dembo, S C Eisenstat, and T Steihaug, “Inexact Newton methods, ” SIAM Journal on Numerical Analysis, vol 19, no 2, pp 400–408, 1982 26 J S Pang, “Inexact Newton methods for the nonlinear complementarity problem,” Mathematical... ξi , 1 4.14 i 0 are finite The rest of the proof is similar to that of Theorem 3.3 and is thus omitted here 5 Computational Results In this section, we present some numerical results for the proposed self-adaptive implicit methods Our main interests are two folds: the first one is to compare the proposed method with He’s method 11 in solving a simple nonlinear problem, showing the numerical advantage; . Inequalities and Applications Volume 2009, Article ID 458134, 20 pages doi:10.1155/2009/458134 Research Article Self-Adaptive Implicit Methods for Monotone Variant Variational Inequalities Zhili Ge and. u 0 . 6. Conclusions In this paper, we proposed a self-adaptive implicit method for solving monotone variant variational inequalities. The proposed self-adaptive adjusting rule avoids the difficult. solutions of variational inequalities,” Journal of Inequalities and Applications, vol. 2008, Article ID 284345, 12 pages, 2008. 11 B. S. He, “Inexact implicit methods for monotone general variational