finite step relaxed hybrid steepest descent methods for variational inequalities

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 598632, 13 pages doi:10.1155/2008/598632 Research Article Finite-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities Yen-Cherng Lin Department of Occupational Safety and Health, General Education Center, China Medical University, Taichung 404, Taiwan Correspondence should be addressed to Yen-Cherng Lin, yclin@mail.cmu.edu.tw Received 22 August 2007; Revised January 2008; Accepted 13 March 2008 Recommended by Jong Kim The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space was studied A new finite-step relaxed hybrid steepest-descent method for this class of variational inequalities was introduced Strong convergence of this method was established under suitable assumptions imposed on the algorithm parameters Copyright q 2008 Yen-Cherng Lin 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 Introduction Let H be a real Hilbert space with inner product ·, · and norm · Let C be a nonempty closed convex subset of H, and let F : C → H be an operator The classical variational inequality problem: find u∗ ∈ C such that F u∗ , v − u∗ ≥ 0, VI F, C ∀v ∈ C, 1.1 was initially studied by Kinderlehrer and Stampacchia It is also known that the VI F, C is equivalent to the fixed-point equation u∗ PC u∗ − μF u∗ , 1.2 where PC is the nearest point projection from H onto C, that is, PC x argminy∈C x − y for each x ∈ H and where μ > is an arbitrarily fixed constant If F is strongly monotone and Lipschitzian on C and μ > is small enough, then the mapping determined by the right-hand side of this equation is a contraction Hence the Banach contraction principle guarantees that the Picard iterates converge in norm to the unique solution of the VI F, C Such a method is called the projection method However, Zeng and Yao point out that the fixed-point Journal of Inequalities and Applications equation involves the projection PC which may not be easy to compute due to the complexity of the convex set C To reduce the complexity problem probably caused by the projection PC , a class of hybrid steepest-descent methods for solving VI F, C has been introduced and studied recently by many authors see, e.g., 3, Zeng and Yao have established the method of two-step relaxed hybrid steepest-descent for variational inequalities A natural arising problem is whether there exists a general relaxed hybrid steepest-descent algorithm that is more than two steps for finding approximate solutions of VI F, C or not Motivated and inspired by the recent research work in this direction, we introduce the following finite step relaxed hybrid steepest-descent algorithm for finding approximate solutions of VI F, C and aim to unify the convergence results of this kind of methods k k Algorithm 1.1 Let {αn } ⊂ 0, , {λn } ⊂ 0, , for k 1, 2, , m, and take fixed numbers t k ∈ 0, 2η/κ2 , k 1, 2, , m Starting with arbitrarily chosen initial points u0 ∈ H, compute the k sequences {un } such that un 1 1 − αn T un − λn t F T un 1 − αn T un − λn t F T un 1 − αn T un − λn t F T un αn un αn un αn un un un 2 , 3 , 4 , 1.3 m m un αn un m − αn m T un − λn t m F T un 1 We will prove a strong convergence result for Algorithm 1.1 under suitable restrictions imposed on the parameters Preliminaries The following lemmas will be used for proving the main result of the paper in next section Lemma 2.1 see Let {sn } be a sequence of nonnegative real numbers satisfying the inequality sn ≤ − αn sn αn τn γn , ∀n ≥ 0, 2.1 where {αn }, {τn }, and {γn } satisfy the following conditions: i {αn } ⊂ 0, , ∞ n αn ∞, or equivalently, ∞ n − αn 0; ii lim supn→∞ τn ≤ 0; iii {γn } ⊂ 0, ∞ , Then limn→∞ sn ∞ n γn < ∞ Lemma 2.2 see Demiclosedness principle: assume that T is a nonexpansive self-mapping on a nonempty closed convex subset C of a Hilbert space H If T has a fixed point, then I − T is demiclosed; that is, whenever {xn } is a sequence in C weakly converging to some x ∈ C and the sequence { I −T xn } strongly converges to some y ∈ H, it follows that I − T x y Here I is the identity operator of H Yen-Cherng Lin The following lemma is an immediate consequence of an inner product Lemma 2.3 In a real Hilbert space H, there holds the inequality x y ≤ x 2 y, x y , ∀x, y ∈ H 2.2 Lemma 2.4 Let C be a nonempty closed convex subset of H For any x, y ∈ H and z ∈ C, the following statements hold: i PC x − x, z − PC x ≥ 0; ii P C x − PC y ≤ x−y − PC x − x y − PC y Convergence theorem Let H be a real Hilbert space and let C be a nonempty closed convex subset of H Let F : C → H be an operator such that for some constants κ, η > 0, F is κ-Lipschitzian and η-strongly monotone on C; that is, F satisfies the conditions Fx − Fy ≤ κ x − y , ∀x, y ∈ C, Fx − Fy, x − y ≥ η x − y , ∀x, y ∈ C, 3.1 respectively Since F is η-strongly monotone, the variational inequality problem VI F, C has a unique solution u∗ ∈ C see, e.g., Assume that T : H → H is a nonexpansive mapping with the fixed points set Fix T C Note that obviously Fix PC C For any given numbers λ ∈ 0, and μ ∈ 0, 2η/κ2 , we define the mapping Tμλ : H → H by Tμλ x : T x − λμF T x , ∀x ∈ H Lemma 3.1 see Let Tμλ be a contraction provided that < λ < and < μ < 2η/κ2 Indeed, Tμλ x − Tμλ y ≤ − λτ x − y , where τ 1− ∀x, y ∈ H, 3.2 − μ 2η − μκ2 ∈ 0, We now state and prove the main result of this paper Theorem 3.2 Let H be a real Hilbert space and let C be a nonempty closed convex subset of H Let F : C → H be an operator such that for some constants κ, η > 0, F is κ-Lipschitzian and ηstrongly monotone on C Assume that T : H → H is a nonexpansive mapping with the fixed points set k k Fix T C, the real sequences {αn }, {λn }, for k 1, 2, , m, in Algorithm 1.1 satisfy the following conditions: i k ∞ n |αn ii limn→∞ αn k − αn−1 | < ∞, for k and 1, 2, , m; k limn→∞ αn 1, for k 2, 3, , m; Journal of Inequalities and Applications 1 iii limn→∞ λn 0, limn→∞ λn /λn k iv λn ≥ max{λn : k ∞ n λn 1, ∞; 2, 3, , m}, for all n ≥ k Then the sequences {un } generated by Algorithm 1.1 converge strongly to u∗ which is the unique solution of the VI F, C Proof Since F is η-strongly monotone, by , the VI F, C has the unique solution u∗ ∈ C Next we divide the rest of the proof into several steps k 1, 2, , m Indeed, let us denote that Ttλ u∗ Step Let {un } is bounded for each k λtF T u∗ , then we have un − u∗ 1 ≤ αn 1− un − u∗ 2 un − 1 − λn τ 3.3 λ Tt 1n 1 λ Tt 1n ∗ u un − u∗ ∗ ∗ u −u λn t F u∗ , λ Tt 2n un − u∗ − αn ∗ 1− −u λ λ αn 1− λn τ k un − u∗ − αn k − αn λ k un − u∗ − αn k un − u∗ − αn k un − u∗ − αn ≤ αn ≤ αn 1 k λn t −u − αn λ k k Tt kn un k − λn τ k λn t F u∗ 3.4 ∗ F u , 2, 3, , m − 1, − u∗ Tt kn un k ≤ αn λ Tt 2n u∗ − u∗ ∗ un − t 2η − t κ2 ∈ 0, , and for k αn un Tt 2n un − Tt 2n u∗ − αn un − u∗ k 1 un ≤ αn un − u∗ 1 αn 1− λ Tt 1n αn − αn un − u∗ 2 λ Tt 1n un − u∗ − t 2η − t κ2 ∈ 0, Moreover, we also have ≤ αn where τ 1− −u αn un ≤ un − u∗ 1 − αn ∗ un ≤ αn λ Tt 1n un − u∗ 1 − αn un − u∗ ≤ αn where τ αn un T u∗ − k k un λ k − u∗ k λ − Tt kn u∗ k k un k Tt kn u∗ − u∗ k − u∗ k − αn λn t k F u∗ k λn t k F u∗ , 3.5 Yen-Cherng Lin where τ m k − t k 2η − t k κ2 ∈ 0, , and 1− m − u∗ un m λ m αn un Tt mn un − u∗ − αn m un − u∗ − αn m un − u∗ − αn ≤ αn ≤ αn λ m m − λn τ m λn t m F u∗ λ Tt mn un − Tt mn u∗ m ≤ un − u∗ m m m λ m Tt mn u∗ − u∗ m un − u∗ λn t m F u∗ , 3.6 where τ m − t m 2η − t m κ2 ∈ 0, 1− Thus we obtain m−1 − u∗ un m−1 un − u∗ − αn m−1 un − u∗ − αn un − u∗ m−1 m ≤ αn ≤ αn m−1 m−1 ≤ un − u∗ 1 − αn k m un m−1 − u∗ λn m m−1 tm m j k 1 − αn max λn k≤j≤m m−1 λn t m F u∗ max λn , λn un − u∗ ≤ un − u∗ t m−1 F u∗ t m−1 tj λn F u∗ F u∗ t m−1 F u∗ , , j k 3.7 for k 2, 3, , m − In particular, m j un − u∗ ≤ un − u∗ − αn max λn 2≤j≤m tj F u∗ 3.8 j Hence, substituting 3.8 in 3.3 and by condition iv , we obtain un − u∗ un − u∗ − αn un − u∗ − αn ≤ αn ≤ αn 1 1 u2 − u∗ 1 un − u∗ 1 − λn τ 1 − λn τ j max {λn } 2≤j≤m ≤ αn un − u∗ 1 − αn 1 − λn τ λn t F u∗ 1 m 3.9 t j F u∗ j un − u∗ λn t F u∗ m j max λn 1≤j≤m t j F u∗ j By induction, it is easy to see that un − u∗ ≤ M, ∀n ≥ 0, 3.10 Journal of Inequalities and Applications m j j 1t max{ u0 − u∗ , where M u1 − u∗ ≤ α0 u0 − u∗ 1 1 − λ1 τ 1 − λ1 τ − α0 F u∗ } Indeed, for n 1 − α0 ≤ α0 M /τ 1 λ1 τ M M m j u0 − u∗ 0, from 3.9 we obtain max λ1 1≤j≤m tj F u∗ j M 3.11 1 Suppose that un − u∗ ≤ M, for n ≥ We want to claim that un un ∗ ≤ αn 1−u un − u∗ 1 1 ≤ αn M 1 − αn − λn τ 1 − λn τ − αn − u∗ ≤ M Indeed, un − u∗ 1 λn τ M M m λn t j F u∗ j M 3.12 m m Therefore, we have un − u∗ ≤ M, for all n ≥ 0, and un − u∗ ≤ M λn τ for all n ≥ In this case, from 3.8 , it follows that j k un − u∗ ≤ M max λn k≤j≤m 1 τ M≤ τ M, ∀n ≥ 0, ∀k M≤ τ 2, 3, , m − 1 M, 3.13 k Step Let un − T un → 0, n → ∞ Indeed by Step 1, {un } is bounded for ≤ k ≤ m and k k 0, so are {T un } and {F T un } for ≤ k ≤ m Thus from the conditions that limn→∞ αn k limn→∞ αn 1, for k 2, 3, , m and limn→∞ λn 0, we have, for k 2, , m, k un − un k k αn un − αn k − − αn k ≤ − αn un k T un k k k 1 − αn T un k T un − un k 3.14 k − λn t k F T un k 1 − αn un k − λn t k F T un k k λn t k F T un −→ and so un 1 − T un 1 αn un αn − αn 1 un − T un un − T un un − T un ≤ αn ≤ αn 2 T un − λn t F T un T un − T u1 − αn 1 − αn 1 un − un − T un T un − T un 1 − λn t F T un 1 − αn λn t F T un 3.15 λn t F T un −→ as n −→ ∞ Yen-Cherng Lin Step Let un m − un → 0, as n → ∞ Indeed, we observe that m un − un−1 m m m ≤ αn 1 − αn m un − un−1 λ m m m αn λ m m λ m Tt mn un − Tt mn un−1 m − λn τ λ m λ m 1 m m m m λn − αn m ≤ un − un−1 − αn m un−1 m m m 1 un − un−1 · t m F T un−1 − − αn−1 λn m m m λn − αn−1 · λn m un − un−1 m · t m F T un−1 − − αn−1 λn m − αn m 1 − αn m λn τ m 1 − − αn αn m m − αn − αn−1 · un−1 − αn−1 · T un−1 1 m m m αn m − αn−1 · un−1 un − un−1 ≤ αn Tt mn un − − αn−1 Tt mn un−1 Tt mn un−1 − − αn−1 Tt mn un−1 − αn m αn m λ m αn un − αn−1 un−1 m m m − αn−1 · un−1 T un−1 · t m F T un−1 − − αn−1 λn T un−1 αn , 3.16 and, for ≤ k ≤ m − 1, k k un − un−1 k k 1 k αn un − αn−1 un−1 k ≤ αn k un − un−1 λ k − αn k k k k k k k λ k k k λ k k k λ k Tt kn un k − λn τ λ k k k − Tt kn un−1 k 1 k k αn − αn−1 · k − λn τ k k k un un−1 k − un−1 k − αn λn − αn k un − un−1 − αn k − − αn−1 Tt kn un−1 − αn αn − αn−1 · un−1 k k 1 k αn − αn−1 · T un−1 αn k αn − αn−1 · un−1 un − un−1 k λ Tt kn un Tt kn un−1 − − αn−1 Tt kn un−1 ≤ αn − αn k k − − αn−1 λn k k un k − un−1 k · t k F T un−1 k T un−1 k − αn k λn k k − − αn−1 λn k · t k F T un−1 , 3.17 Journal of Inequalities and Applications m m m − un−1 ≤ un − un−1 un αn m T un−1 m−1 m−1 un m−1 m−1 m−1 m λn m−1 m m−1 m−1 − − αn−1 t m F T un−1 m λn , t m FT un−1 m t m−1 FT un−1 3.18 m T un−1 1 un−1 − αn−1 m − − αn−1 λn λn un−1 − αn−1 m αn m m 1 − αn − αn un−1 − − αn−1 λn λn m αn m 1 − αn ≤ un − un−1 − un−1 m − αn−1 · T un−1 , 2 m−1 un − un−1 ≤ un − un−1 m−1 k k k − αn λn k k k un−1 k k − − αn−1 λn t k FT un−1 k 1 αn − αn−1 k m T un−1 m αn 1 un−1 − αn−1 T un−1 3.19 Hence it follows from the above inequalities 3.17 – 3.19 that un − un 1 1 1 1 ≤ αn 1 − αn λ 1 1 1 1 λ λ Tt 1n un − Tt 1n un−1 1 − λn τ λn k un−1 1 k T un−1 − − αn−1 λn 1 − − αn−1 λn un − un−1 1 − αn λn − αn 1 1 αn − αn−1 · − λn τ − αn λ un − un−1 − αn 1 − αn αn − αn−1 · un−1 1 λ 1 αn − αn−1 · T un−1 αn αn − αn−1 · un−1 un − un−1 1 Tt 1n un − − αn−1 Tt 1n un−1 Tt 1n un−1 − − αn−1 Tt 1n un−1 ≤ αn − αn un − un−1 λ αn un − αn−1 un−1 · t F T un−1 2 un − un−1 · t F T un−1 3.20 Yen-Cherng Lin Let us substitute 3.19 into 3.20 , then we have un − un 1 ≤ αn 1 − αn m−1 k k − αn m 1 − αn λn 1 − − αn 1 m−1 un − un−1 k λn k m − λn τ k k m k k 1 un−1 T un−1 k − − αn−1 λn t k · FT un−1 m m − − αn−1 λn t m FT un−1 λn τ k αn − αn−1 · 1 1 un − un−1 m αn − αn λn τ 1 νn un−1 − αn−1 T un−1 δn , 3.21 where δn m−1 k νn k k αn − αn−1 · 1 αn 1− k 1 un−1 m m 1 − αn λn τ m T un−1 m−1 λn αn m m m − − αn−1 λn k − αn k k λn un−1 T un−1 , t m FT un−1 k 1 − αn−1 · 3.22 k k − − αn−1 λn t k FT un−1 We put ξ sup un k FT un sup M :n≥0 m u∗ sup : n ≥ 0, k tk F u∗ k T un : n ≥ 0, k 1, 2, , m 1, 2, , m , 3.23 ξ k k m k |αn Then δn ≤ 2M m νn ≤ tk k × m k M k − αn−1 | → 0, as n → ∞, and 1− k − αn αn λn τ k λn 3.24 k k − − αn−1 λn 1 − αn λn 1 − − αn−1 λn 10 Journal of Inequalities and Applications ∞ n δn From ii – iv , we obtain νn → as n → ∞ Furthermore, from i , 1 we deduce that un − un → as n → ∞ 1 Step Let un − T un < ∞ By Lemma 2.1, → as n → ∞ From Steps and 3, we have 1 un − T un ≤ un 1 − un un 1 − T un −→ 3.25 as n → ∞ k Step Let lim supn→∞ −F u∗ , T un − u∗ ≤ 0, for k of {T un } such that lim sup − F u∗ , T un − u∗ 2, 3, , m Let {T uni } be a subsequence lim − F u∗ , T uni − u∗ 1 3.26 i→∞ n→∞ Without loss of generality, we assume that T uni → u weakly for some u ∈ H By Step 4, we derive uni → u weakly But by Lemma 2.2 and Step 4, we have u ∈ Fix T C Since ∗ u is the unique solution of the VI F, C , we obtain lim sup − F u∗ , T un − u∗ n→∞ − F u∗ , u − u∗ ≤ 3.27 −→ 0, 3.28 From the proof of Step 2, k T un − T un for k ≤ un − un as n −→ ∞, 2, 3, , m Then k lim sup − F u∗ , T un − u∗ n→∞ lim sup n→∞ k − F u∗ , T un − T un k ≤ lim sup − F u∗ , T un − T un n→∞ lim sup − F n→∞ u∗ , T un − F u∗ , T un − u∗ lim sup − F u∗ , T un − u∗ n→∞ − u∗ ≤ 0, 3.29 for k 2, 3, , m Yen-Cherng Lin 11 k Step Let un → u∗ in norm and so does {un } for k and 3.7 we get − u∗ un un − u∗ 1 αn un − u∗ − αn un − u∗ − αn un − u∗ − αn ≤ αn ≤ αn ≤ αn −u 1 λ λ λ Tt 1n un − Tt 1n u∗ 2 1 − λn τ 1 − λn τ 2 − αn 1 1 − λn τ 1 un ∗ −u 1− 2 αn m max j m j λn t j ∗ F u j − u∗ 2 m j max {λn } 2≤j≤m λ − u∗ un − u∗ 1 − λn τ − αn − αn 1− λn τ 1 2 2 2 − αn λ 2 Tt 1n u∗ − u∗ , Tt 1n un − u∗ un − u∗ 1 λ Tt 1n u∗ − u∗ − F u∗ , T un − λn t F T un λn t 1 tj F u∗ un − u∗ j 2 j m max {λn } 2≤j≤m tj F u∗ j − F u∗ , T un − u∗ − λn t F T un 1 ≤ − − αn 1− 1 αn 1− − αn 2t λn λ Tt 1n un − Tt 1n u∗ 2 − αn ≤ αn 2 1 ∗ un 2 − F u∗ , T un − λn t F T un 2t λn Tt 1n un − u∗ − αn λ αn Tt 1n un − u∗ un − u∗ λ 1 − αn ≤ αn 2, 3, , m Indeed using Lemma 2.3 αn λn τ 1− αn un −u∗ 1− 2 1−αn λn τ 1−λn τ 1−αn j max {λ } 2≤j≤m n m j − F u∗ , T un − u∗ − λn t F T un 2λn t 1 ≤ − − αn λn τ 1 un − u∗ 2 1 − αn λn τ 1 t M2 m j max λn 2≤j≤m j j tj M2 12 Journal of Inequalities and Applications × τ 1 − αn 1 − λn τ − αn 2 − αn 1 − λn τ j ≤ 1− 1− λn τ 1 un j τ τ 2 − αn 1 1 − αn λn τ τ max {λn } 2≤j≤m 1 − λn τ − αn 1 m tj M2 j 2 m j 1 − αn M2 −u − λn τ λn ∗ 2 m j j 2t max2≤j≤m λn − F u∗ , T un − u∗ − λn t F T un 2t M2 λn τ αn m j j 2t max2≤j≤m {λn } τ × − F u∗ , T un − u∗ − λn t F T un 2t tj M2 j 3.30 k From ii , iii , and Step 5, we have limn→∞ αn limn→∞ λn 0, for k 1, 2, , m and k 2 ∗ ∗ limn→∞ αn 1, for k 2, , m, lim supn→∞ −F u , T un −u ≤ 0, and {F T un } is bounded; by Lemma 2.4, we conclude that lim sup τ n→∞ − αn τ 1 − λn τ ≤0 2t n→∞ τ 0 1 j max2≤j≤m {λn } 1− m j j 2t 2 m j j 2t M2 M2 · − F u∗ , T un − u∗ αn 1 − λn τ − αn τ ≤ lim sup 2 − αn − F u∗ , T un − u∗ − λn t F T un 2t lim sup n→∞ t1 τ 1− λn αn · − F u∗ , −F T un 3.31 k Consequently from Lemma 2.1, we obtain un − u∗ → and hence it follows from un − k un → 0, for k 2, 3, , m, that un − u∗ → 0, for k 2, 3, , m Acknowledgment This research was partially supported by Grant no NSC95-2115-M-039-001- from the National Science Council of Taiwan References D Kinderlehrer and G Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol 88 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1980 Yen-Cherng Lin 13 L C Zeng and J.-C Yao, “Two step relaxed hybrid steepest-descent methods for variational inequalities,” to appear in Applied Mathematics and Mechanics I Yamada, “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, D Butnariu, Y Censor, and S Reich, Eds., vol of Studies in Computational Mathematics, pp 473–504, North-Holland, Amsterdam, The Netherlands, 2001 L C Zeng, N C Wong, and J.-C Yao, “Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities,” Journal of Optimization Theory and Applications, vol 132, no 1, pp 51–69, 2007 H.-K Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol 66, no 1, pp 240–256, 2002 K Goebel and W A Kirk, Topics in Metric Fixed Point Theory, vol 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990 J.-C Yao, “Variational inequalities with generalized monotone operators,” Mathematics of Operations Research, vol 19, no 3, pp 691–705, 1994 ... hybrid steepest- descent for variational inequalities A natural arising problem is whether there exists a general relaxed hybrid steepest- descent algorithm that is more than two steps for finding... step relaxed hybrid steepest- descent methods for variational inequalities, ” to appear in Applied Mathematics and Mechanics I Yamada, “The hybrid steepest- descent method for the variational inequality... introduce the following finite step relaxed hybrid steepest- descent algorithm for finding approximate solutions of VI F, C and aim to unify the convergence results of this kind of methods k k Algorithm