Bounded perturbation resilience of the viscosity algorithm

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1 Dong et al. Jornal of Ineqalities and Applications 2016) 2016:299 DOI /s R E S E A R C H Open Access Bonded pertrbation resilience of the viscosity algorithm Qiao-Li Dong *, Jing Zhao and Songnian He * Correspondence: dongql@lsec.cc.ac.cn College of Science, Civil Aviation University of China, Tianjin, , China Abstract In this article, we investigate the bonded pertrbation resilience of the viscosity algorithm and propose the speriorized version of the viscosity algorithm. The convergence of the proposed algorithm is analyzed for a nonexpansive mapping. A modified viscosity algorithm and its convergence is presented. Keywords: bonded pertrbation resilience; constrained minimization; the viscosity algorithm; speriorization 1 Introdction and preliminaries Let H be a real Hilbert space and C be a nonempty closed and convex sbset of H.Aselfmapping f : C C is called a contraction with ρ 0, 1) if f x) f y) ρ x y, x, y C. A mapping T : C C is called nonexpansive if Tx Ty x y, x, y C. Denote by FixT) thesetoffixedpointsoft, i.e., FixT)={x C : x = Tx}. Throghot this article we assme that FixT). The fixed point problems for nonexpansive mappings [1 4] captre varios applications in diversified areas, sch as convex feasibility problems, convex optimization problems, problems of finding the zeros of monotone operators, and monotone variational ineqalities see [1, 5] and references therein). Many iteration algorithms were introdced to approximate a fixed point of a nonexpansive mapping in a Hilbert space or a Banach space for example, see [6 14]). The iteration algorithms are sally divided into two kinds. One is the algorithms with weak convergence, sch as the Mann iteration algorithm [15] andtheishikawaiteration algorithm [16]. The other is the algorithms with strong convergence, sch as the Halpern iteration algorithm [17], hybrid algorithms [18], and the shrinking projection algorithm [19]. Dong et al This article is distribted nder the terms of the Creative Commons Attribtion 4.0 International License which permits nrestricted se, distribtion, and reprodction in any medim, provided yo give appropriate credit to the original athors) and the sorce, provide a link to the Creative Commons license, and indicate if changes were made.

2 Dongetal. Jornal of Ineqalities and Applications 2016) 2016:299 Page 2 of 12 As an extension to Halpern s iteration process, Modafi [20] introdced the viscosity algorithm which is defined as follows: x k+1 = α k f x k) +1 α k )Tx k 1) where {α k } is a seqence in the interval [0, 1]. For a contraction f and a nonexpansive mapping T, Modafi proved the strong convergence of {x k } provided that {α k } satisfies the following conditions: C1) α k 0 k ); C2) + k=1 α k =+ ; C3) + k=1 α α k+1 α k <+ or lim k k =1. α k+1 X [21] stdied the viscosity algorithm in the setting of Banach spaces and obtained the strong convergence theorems. Recently, Yang and He [22] proposed the so-called general alternative reglarization algorithm: x k+1 = T α k f x k) +1 α k )x k), 2) where {α k } is a seqence in the interval [0, 1]. Actally, the general alternative reglarization algorithm is a variant of the viscosity algorithm 1). Define T k : C C by T k x = T α k f x)+1 α k )x ). 3) Then the viscosity algorithm 2)can be rewritten as x k+1 = T k x k ). 4) Under the conditions C1)-C3), Yang and He [22] showed the strong convergence of {x k } provided that f is a Lipschitzian and strongly psedo-contractive mapping. It is obvios that if f is a contraction, then f is a Lipschitzian and strongly psedo-contractive mapping. So, from Theorem 3.1 in [22], we can get the following reslts. Theorem 1.1 Let C be a nonempty closed convex sbset of a Hilbert space H. Let T : C C be a nonexpansive mapping with FixT) and let f : C C be a contraction. If {α k } 0, 1) satisfies the conditions C1)-C3). Then the seqence {x k } generated by 4) converges strongly to a fixed point x of T, where x is the niqe soltion of the variational ineqality x f x ), x x 0, x FixT). The speriorization methodology was first proposed by Btnari et al. [23] andfor- mlated over general problem strctres in [24]. Recently, there are increasing interests in stdying of the speriorization methodology see [25 30] and the references therein). Davidi et al. [25] analyzed pertrbation resilience of the class of block-iterative projection BIP) methods. Jin et al. [30] stdied the bonded pertrbation resilience of projected scaled gradient methods PSG) and applied the speriorization methodology to the PSG. Censor et al. [26] introdced the speriorized version of the dynamic string-averaging

3 Dongetal. Jornal of Ineqalities and Applications 2016) 2016:299 Page 3 of 12 projection algorithm and revealed new information as regards the mathematical behavior of the speriorization methodology. Herman and Davidi [27] stdied the advantages of speriorization for image reconstrction from a small nmber of projections. Nikazad et al. [28] proposed two acceleration schemes based on BIP methods. In [29], the athors investigated total variation speriorization schemes in proton compted tomography image reconstrction. Or aim in this article is to investigate the speriorization and the bonded pertrbation resilience of the viscosity algorithm and constrct algorithms based on them. Next, we list two lemmas needed in the proof of the main reslts. Lemma 1.1 [6]) Assme {a n } is a seqence of nonnegative real nmbers sch that a n+1 1 γ n )a n + γ n δ n + β n, where {γ n } 0, 1) and {β n } 0, ) and {δ n } R satisfies 1) n=1 γ n = ; 2) lim sp n δ n 0 or n=1 γ n δ n < ; 3) n=1 β n <. Then lim n a n =0. Lemma 1.2 see [1], Corollary 4.18) Let D H be nonempty closed and convex, T : D H be nonexpansive and let {x n } be a seqence in D and x H sch that x n xandtx n x n 0 as n +. Then x FixT). 2 The speriorization methodology Consider some mathematically formlated problem denoted by Problem P,thesetofsoltions of which is denoted by SOLP). The speriorization methodology of [24, 26, 27]is intended for constrained minimization problems: minimize { φx) x P }, 5) where φ : R J R is an objective fnction and P R J is the soltion set P = SOLP)of aproblemp. Here, we assme P = SOLP) throghot this paper. The speriorization methodology strives not to solve 5), bt rather the task is to find apointin P which is sperior, i.e., has a lower, bt not necessarily minimal, vale of the objective fnction φ. From Theorem 1.1, the viscosity algorithm x k+1 = T α k f x k) +1 α k )x k), converges to a fixed point x of a nonexpansive mapping T, which satisfies the following variational ineqality: x f x ), x x 0, x FixT). It is well known that the constrained optimization problem min x C φx)

4 Dongetal. Jornal of Ineqalities and Applications 2016) 2016:299 Page 4 of 12 is eqivalent to a variational ineqality as follows: find x C, schthat φ x ), x x 0, x C, provided that φ is differentiable. Hence, if the fnction f is specially chosen, the viscosity algorithm converges to a soltion of a constrained optimization problem: min x FixT) φx). This motivates s to investigate the viscosity algorithm by sing speriorization methodology. In the following sections, we first investigate the bonded pertrbation resilience of the viscosity algorithm and se the bonded pertrbation resilience to introdce an inertial viscosity algorithm. Then we present the speriorized version of the viscosity algorithm and analyze its convergence. Finally, we give a modified viscosity algorithm. 3 The bonded pertrbation resilience of the viscosity algorithm In this section, we discss the bonded pertrbation resilience of the viscosity algorithm 2). First, we present the definition of bonded pertrbation resilience which is essential in the speriorization algorithm. Definition 3.1 [26]) Given a problem P, an algorithmic operator A : H H is said to be bonded pertrbations resilient iff the following is tre: if the seqence {x k } k=0, generated by x k+1 = Ax k ), for all k 0, converges to a soltion of P for all x 0 H,thenanyseqence {y k } k=0 of points in H that is generated by yk+1 = Ay k +α k v k ), for all k 0, also converges to asoltionofp provided that, for all k 0, α k v k are bonded pertrbations, meaning that α k 0 for all k 0schthat k=0 α k < and sch that the seqence {v k } k=0 is bonded. Since Theorem 1.1 holds, we only need to prove the following reslts for showing the bonded pertrbation resilience of the viscosity algorithm. Theorem 3.1 Let T : H H be a nonexpansive mapping with FixT) and let f : H H be a contraction. Let {α k } be a seqence of nonnegative nmbers satisfies C1)-C3) and let {β k } be a seqence of nonnegative nmbers sch that β k α k and k=1 β k <. Let {v k } H be a norm-bonded seqence. Let {T k } be defined by 3). Then any seqence {y k } generated by the iterative formla y k+1 = T k y k + β k v k) 6) converges strongly to a fixed point y of T, where y is the niqe soltion of the variational ineqality y f y ), y y 0, y FixT). 7)

5 Dongetal. Jornal of Ineqalities and Applications 2016) 2016:299 Page 5 of 12 Proof First we show that {y k } is bonded. Let k = y k + β k v k. 8) Then, for any p FixT), we have k p = y k + β k v k p y k p + βk v k y k p + βk M 1, 9) where M 1 := sp k N { v k }.From6)and8), it follows that y k+1 p = Tk k ) p By indction, α k f k) +1 α k ) k p ) α k f k ) p +1 αk ) k p α k f k ) f p) + f p) p ) +1 αk ) k p α k ρ k p + f p) p ) +1 αk ) k p = 1 1 ρ)α k ) k p + αk f p) p ) 1 1 ρ)α k y k p + αkf p) p + βk M 1 = ) 1 1 ρ)α k y k p + α k f p) p + β ) k M 1 α k ) 1 1 ρ)α k y k p + α k f p) p ) + M 1 { y max k p 1 ) }, f p) p + M1. 1 ρ y k p { y max 0 p 1 ) }, f p) p + M1, k 0, 1 ρ and {y k } is bonded, so are {f k )} and { k }. We now show that y k Ty k 0. 10) Indeed we have y k+1 Ty k = T k k ) Ty k = T α k f k) +1 α k ) k) Ty k αk f k) +1 α k ) k y k α k f k ) y k +1 αk ) k y k = α k f k ) y k + βk v k α k + β k )M 2 0, 11)

6 Dongetal. Jornal of Ineqalities and Applications 2016) 2016:299 Page 6 of 12 where M 2 := max{sp k N { f k ) + y k }, M 1 }.By8), we have k k 1 y k y k 1 + βk v k + βk 1 v k 1. From 6), it followsthat y k+1 y k = T k k T k 1 k 1 α k f k) +1 α k ) k α k 1 f k 1) 1 α k 1 ) k 1 = 1 α k ) k k 1) +α k α k 1 ) f k 1) k 1) + α k f k ) f k 1)) 1 1 ρ)α k ) k k 1 + α k α k 1 f k 1) k ρ)α k ) y k y k 1 + α k α k 1 M 3 + β k v k + β k 1 v k 1, where M 3 := sp k N { f k ) + k }. By Lemma 1.1,weobtain y k+1 y k 0. 12) Combining 11)and12), we get 10). We next show lim sp y y k, y f y ) 0, k where y satisfies 7). Indeed take a sbseqence {y k l} of {y k } sch that lim sp y y k, y f y ) = lim y y k l, y f y ). k l We may assme that y k l ȳ. It follows from Lemma 1.2 and 10) thatȳ FixT). Hence by 7)weobtain lim sp y y k, y f y ) = y ȳ, y f y ) 0, k as reqired. Finally we show that y k y.using9), we have y k+1 y 2 1 αk ) k y ) + α k f k ) y ) 2 =1 α k ) 2 k y 2 + αk 2 f k ) y 2 +2α k 1 α k ) k y, f k) y =1 α k ) 2 k y 2 + αk 2 f k) y 2 +2α k 1 α k ) k y, f k) f y ) +2α k 1 α k ) k y, f y ) y 1 2α k + α 2 k +2ρα k1 α k ) ) k y 2

7 Dongetal. Jornal of Ineqalities and Applications 2016) 2016:299 Page 7 of 12 where and + α k [ αk f k ) y α k ) k y, f y ) y ] [1 ᾱ k ] [ y k y 2 +2β k y k y M1 + β 2 k M2 1 ] + ᾱk δ k + β k M 4 [1 ᾱ k ] y k y 2 + ᾱ k δ k + β k M 4, ᾱ k = α k 2 αk 2ρ1 α k ) ), δ k = α k f k ) y α k ) k y, f y ) y, 2 α k 2ρ1 α k ) { M 4 = sp 2 y k y M 1 + β k M 2 } 1, k N where M 4 <+ from the bondedness of {y k }.Itiseasilyseenthatᾱ k 0, k=1 ᾱk =, and lim sp k δ k 0. By Lemma 1.1,weobtainthereslts. The inertial-type algorithms originate from the heavy ball method an implicit discretization) of the second-order dynamical systems in time [31, 32], the main featre of which is that the next iteration is defined by making se of the previos two iterates. The inertial-type algorithms speed p the original algorithm withot the inertial extrapolation. A classical inertial-type algorithm is the well-known FISTA, which has better global rate of convergence than that ofthe ISTA see [33, 34]). Recently there are increasing interests in stdying inertial-type algorithms, e.g., inertial extragradient methods [35], the inertial Mann iteration algorithm [36, 37], and the inertial ADMM [38]. Combining the inertial-type algorithm and the viscosity algorithm, we constrct the following inertial viscosity algorithm: { w k = x k + β k v k, 13) x k+1 = Tα k f w k )+1 α k )w k ), where v k = γ k x k x k 1) 14) and { 1 if x k x k 1 >1, γ k = x k x k 1 1 if x k x k ) Theorem 3.2 Let T : H H be a nonexpansive mapping with FixT) and let f : H H be a contraction. Let {α k } be a seqence of nonnegative nmbers satisfies C1)-C3) and let {β k } be a seqence of nonnegative nmbers sch that β k α k and k=1 β k <. Then any seqence {x k } generated by the iterative formla 13) converges strongly to a fixed point y of T, where y is the niqe soltion of the variational ineqality x f x ), x x 0, x FixT).

8 Dongetal. Jornal of Ineqalities and Applications 2016) 2016:299 Page 8 of 12 Proof From 14) and15), it is obvios that v k 1. Ths {v k } is a norm-bonded seqence. Using Theorem 3.1, wegetthereslts. 4 The speriorized version of the viscosity algorithm In this section, we present the speriorized version of the viscosity algorithm and show its convergence. Let φ : X R be a convex continos fnction. Consider the set C min := { x FixT) φx) φy) for all y FixT) }, 16) and assme that C min. Algorithm 4.1 The speriorized version of the viscosity algorithm) 0) Initialization: Let N be a natral nmber and let y 0 H be an arbitrary ser-chosen vector. 1) Iterative step: Given a crrent vector y k,pickann k {1,2,...,N} and start an inner loop of calclations as follows: 1.1) Inner loop initialization: Define y k,0 = y k. 1.2) Inner loop step: Given y k,n, as long as n < N k do as follows: 1.2.1) Pick a 0<β k,n 1, n =0,...,N k 1, in a way that garantees that N k 1 β k,n <. 17) k= ) Let φy k,n ) be the sbgradient set of φ at y k,n and define v k,n as follows: v k,n = { s k,n s k,n if 0/ φy k,n ), 0 if 0 φy k,n ), 18) where s k,n φy k,n ) ) Calclate y k,n+1 = y k,n + β k,n v k,n, 19) andgotostep1.2). 1.3) Exit the inner loop with the vector y k,n k. 1.4) Calclate y k+1 = T k y k,n k ), 20) and go back to step 1). We will present the convergence of the Algorithm 4.1. Theorem 4.1 Let T : H H be a nonexpansive mapping with FixT). Then any seqence {y k } k=0, generated by Algorithm 4.1 converges strongly to a fixed point y of T, where

9 Dongetal. Jornal of Ineqalities and Applications 2016) 2016:299 Page 9 of 12 y is the niqe soltion of the variational ineqality y f y ), y y 0, y FixT). Proof By Theorem 3.1, we only need to show that {y k,n k y k } is bonded. From Algorithm 4.1, aseqence{y k } has the property that for each integer k 1andeachh {1,2,...,N k }, and ths y k,h y k h = y k,j y k,j 1) N k y k,n y k,n 1 N k 1 β k,n, j=1 y k,n k y k N k 1 β k,n. So, by 17), y k,n k y k N k 1 β k,n. k=0 k=0 n=1 The bonded pertrbation resilience secred by Theorem 3.1 garantees the convergence of {y k } to the niqe soltion of 7). 5 A modified viscosity algorithm Algorithm 4.1 can be seen as a nified frame, based on which, some algorithms are constrcted. In this section, we introdce a modified viscosity algorithm by choosing a special fnction φx)onalgorithm4.1. Define a convex fnction φ : H R by φx)= 1 2 x 2 hx), where h : H R is a continos fnction. Then the set C min := { x FixT) φx) φy) for all y FixT) }, eqals { } C min := arg min φx). x FixT) Frthermore, if we assme that h : H R is a differentiable fnction, then C min := { x FixT) x f x ), x x 0 for all x FixT) }, where f x)= hx). It is obvios that φ = I f. From Algorithm 4.1, we constrctthe following algorithm.

10 Dongetal.Jornal of Ineqalities and Applications 2016) 2016:299 Page 10 of 12 Algorithm 5.1 A modified viscosity algorithm) 0) Initialization: Let N be a natral nmber and let y 0 C be an arbitrary ser-chosen vector. 1) Iterative step: Given a crrent vector y k,pickann k {1,2,...,N} and start an inner loop of calclations as follows: 1.1) Inner loop initialization: Define y k,0 = y k. 1.2) Inner loop step: Given y k,n, as long as n < N k do as follows: 1.2.1) Pick a 0 β k,0 1 for N k =1and 0<β k,n 1, n =0,...,N k 1for N k >1, in a way that garantees that N k 1 β k,n <. 21) k= ) Calclate y k,n+1 = β k,n f y k,n) +1 β k,n )y k,n, 22) andgotostep1.2). 1.3) Exit the inner loop with the vector y k,n k. 1.4) Calclate y k+1 = T k y k,n k ), 23) and go back to step 1). Remark 5.1 1) In the modified viscosity algorithm, β k,0 can be taken zero. Frthermore, the modified viscosity algorithm redces to the viscosity algorithm 2)ifN k =1fork N. 2) In the speriorized version of the viscosity algorithm, from the definition of v k,n,it follows that v k,n 1. However, in the modified viscosity algorithm, the bondedness of {v k := y k,n k y k } is not obvios. So, we need to show the bondedness of the {v k }. Theorem 5.1 Let C be a nonempty closed convex sbset of a Hilbert space H and let T : C C be a nonexpansive mapping with FixT). Then any seqence {y k } k=0, generated by Algorithm 5.1, converges strongly to a fixed point y of T, where y is the niqe soltion of the variational ineqality y f y ), y y 0, y FixT). Proof Similar to the proof of Theorem 4.1, we only need to show that the bondedness of {y k,n k y k }.Letp FixT), then, for n =0,1,...,N k 1, it follows that y k,n+1 p 1 α k,n ) y k,n p + α k,n f y k,n) p 1 α k,n ) y k,n p + α k,n f y k,n) f p) + f p) p ) 1 α k,n ) y k,n p + α k,n ρ y k,n p + f p) p ) = ) 1 1 ρ)α k,n y k,n p + α k,n f p) p max{ y k,n p 1, f p) p }. 1 ρ

11 Dongetal.Jornal of Ineqalities and Applications 2016) 2016:299 Page 11 of 12 So, we have y k+1 p y k,n k p max{ y k,nk 1 p 1, f p) p } 1 ρ { y max k p 1, f p) p }, 1 ρ where the first ineqality comes from the nonexpansivity of T. By indction, y k p max{ y 0 p, 1 f p) p }. 1 ρ Ths {y k } and {y k,n }, n =1,...,N k, are bonded, so is {f y k,n )}, n =0,...,N k 1.From22), we have y k,n+1 y k,n = β k,n f y k,n ) y k,n). So, y k,n k y k N k 1 y k,n+1 y k,n N k 1 β k,n f y k,n ) y k,n N k 1 M5 β k,n, where M 5 := sp k N {max 0 n Nk { f y k,n ) + y k,n }}.Ths y k,n k y k M 5 k=0 N k 1 k=0 β k,n. The bonded pertrbation resilience secred by Theorem 3.1 garantees the convergence of {y k } to a point in C. Competing interests The athors declare that they have no competing interests. Athors contribtions All athors contribted eqally to the writing of this paper. All athors read and approved the final manscript. Acknowledgements Spported by National Natral Science Fondation of China No ) and Fndamental Research Fnds for the Central Universities No L006). Received: 12 Agst 2016 Accepted: 10 November 2016 References 1. Baschke, HH, Combettes, PL: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin 2011) 2. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge Stdies in Advanced Mathematics. Cambridge University Press, Cambridge 1990) 3. Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York 1984) 4. Takahashi, W: Nonlinear Fnctional Analysis. Yokohama Pblishers, Yokohama 2000) 5. Baschke, HH, Borwein, JM: On projection algorithms for solving convex feasibility problems. SIAM Rev. 383), ) 6. X, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, ) 7. Dong, QL, Yan, HB: Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping. Fixed Point Theroy Appl. 2015, )

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