Research Article Variant Gradient Projection Methods for the Minimization Problems

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1 Abstract ad Applied Aalysis Volume 2012, Article ID , 16 pages doi: /2012/ Research Article Variat Gradiet Projectio Methods for the Miimizatio Problems Yoghog Yao, 1 Yeog-Cheg Liou, 2 ad Chig-Feg We 3 1 Departmet of Mathematics, Tiaji Polytechic Uiversity, Tiaji , Chia 2 Departmet of Iformatio Maagemet, Cheg Shiu Uiversity, Kaohsiug 833, Taiwa 3 Ceter for Geeral Educatio, Kaohsiug Medical Uiversity, Kaohsiug 807, Taiwa Correspodece should be addressed to Chig-Feg We, cfwe@kmu.edu.tw Received 3 May 2012; Accepted 6 Jue 2012 Academic Editor: Je-Chi Yao Copyright q 2012 Yoghog Yao et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. The gradiet projectio algorithm plays a importat role i solvig costraied covex miimizatio problems. I geeral, the gradiet projectio algorithm has oly weak covergece i ifiite-dimesioal Hilbert spaces. Recetly, H. K. Xu 2011 provided two modified gradiet projectio algorithms which have strog covergece. Motivated by Xu s work, i the preset paper, we suggest three more simpler variat gradiet projectio methods so that strog covergece is guarateed. 1. Itroductio Let H be a real Hilbert space ad C a oempty closed ad covex subset of H. Let f : H R be a real-valued covex fuctio. Now we cosider the followig costraied covex miimizatio problem: mi x C f x. 1.1 Assume that 1.1 is cosistet; that is, it has a solutio ad we use S to deote its solutio set. If f is Fréchet differetiable, the x C solves 1.1 if ad oly if x C satisfies the followig optimality coditio: f x,x x 0, x C, 1.2

2 2 Abstract ad Applied Aalysis where f deotes the gradiet of f.notethat 1.2 ca be rewritte as x x f x ),x x 0, x C. 1.3 This shows that the miimizatio 1.1 is equivalet to the fixed-poit problem: Proj C x γ f x ) x, 1.4 where γ > 0 is a ay costat ad Proj C is the earest poit projectio from H oto C. By usig this relatioship, the gradiet-projectio algorithm is usually applied to solve the miimizatio problem 1.1. This algorithm geerates a sequece {x } through the recursio: x 1 Proj C x γ f x ), 0, 1.5 where the iitial guess x 0 C is chose arbitrarily ad {γ } is a sequece of step sizes which may be chose i differet ways. The gradiet-projectio algorithm 1.5 is a powerful tool for solvig costraied covex optimizatio problems ad has well bee studied i the case of costat stepsizes γ γ for all. The reader ca refer to 1 9. It has recetly bee applied to solve split feasibility problems which fid applicatios i image recostructios ad the itesity modulated radiatio therapy see It is kow 3 that if f has a Lipschitz cotiuous ad strogly mootoe gradiet, the the sequece {x } ca be strogly coverget to a miimizer of f i C. If the gradiet of f is oly assumed to be Lipschitz cotiuous, the {x } ca oly be weakly coverget if H is ifiite dimesioal. This gives aturally rise to a questio. Questio 1. How to appropriately modify the gradiet projectio algorithm so as to have strog covergece? For this purpose, recetly, Xu 18 first itroduced the followig modificatio: x 1 θ h x 1 θ Proj C x γ f x ), 0, 1.6 where the sequeces {θ } 0, 1 ad {γ } 0, satisfy the followig coditios: i lim θ 0, 0 θ ad 0 θ 1 θ < ; ii 0 < lim if γ lim sup γ < 2/L ad 0 γ 1 γ <. Xu 18 proved that the sequece {x } coverges strogly to a miimizer of 1.1. Remark 1.1. Xu s modificatio 1.6 is a covex combiatio of the gradiet-projectio algorithm 1.5 ad a self-mappig h x which is usually referred as a so-called viscosity item.

3 Abstract ad Applied Aalysis 3 I 18, Xu preseted aother modificatio as follows: y Proj C x γ f x ), 0, C { z C : y z x z }, Q {z C : x z, x 0 x 0}, 1.7 x 1 Proj C Q x 0. Cosequetly, Xu 18 proved that Algorithm 1.7 also coverges strogly to x which solves the miimizatio problem 1.1. Remark 1.2. Equatio 1.7 ivolved i additioal projectios which couple the gradiet projectio method 1.5 with the so-called CQ method. It should be poited out that Xu s modificatios 1.6 ad 1.7 are iterestig ad provide us with a directio for solvig 1.1 i ifiite-dimesioal Hilbert spaces. Motivated by Xu s work, i the preset paper, we suggest three variat gradiet projectio methods so that strog covergece is guarateed for solvig 1.1 i ifiitedimesioal Hilbert spaces. Our motivatios are maily i the two respects. Reaso 1. The solutio of the miimizatio problem 1.1 is ot always uique, so that there may be may solutios to the problem. I that case, a special solutio e.g., the miimum orm solutio must be foud from amog cadidate solutios. The miimum orm problem is motivated by the followig least squares solutio to the costraied liear iverse problem: Bx b, x Ω, 1.8 where Ω is a oempty closed covex subset of a real Hilbert space H, B is a bouded liear operator from H to aother real Hilbert space H 1, B is the adjoit of B,adb is a give poit i H 1. The least-squares solutio to 1.8 is the least-orm miimizer of the miimizatio problem: mi Bx b. 1.9 x Ω For some related works, please see Solodov ad Svaiter 19, Goebel ad Kirk 20, ad Martiez-Yaes ad Xu 21. Reaso 2. Projectio methods are used extesively i a variety of methods i optimizatio theory. Apart from theoretical iterest, the mai advatage of projectio methods, which makes them successful i real-world applicatios, is computatioal see I this respect, 1.7 is particularly useful. But we observe that 1.7 ivolves two halfspaces C ad Q.IfthesetsC ad Q are simple eough, the P C ad P Q are easily executed. But C Q may be complicate, so that the projectio P C Q is ot easily executed. This might seriously affect the efficiecy of the method. Hece, it is iterestig that oe ca relax C or Q from 1.7.

4 4 Abstract ad Applied Aalysis I the preset paper, we suggest the followig three methods: x 1 Proj C θ h x 1 θ x γ f x ), 0, x 1 Proj C I γ f ) ProjC 1 θ x, 0, 1.10 y Proj C x γ f x ), 0, C { z C 1 : y z x z }, 1.11 x 1 Proj C x 0. We will show that 1.10 ca be used to fid the miimum orm solutio of the miimizatio problem 1.1,ad 1.11 which is oly ivolved i C also has strog covergece. 2. Prelimiaries Let C be a oempty closed covex subset of a real Hilbert space H. A mappig T : C C is called oexpasive if Tx Ty x y, x, y C. 2.1 Recall that the earest poit or metric projectio from H oto C, deoted Proj C, assigs, to each x H, the uique poit Proj C x C with the property x Proj C x if { x y : y C }. 2.2 It is well kow that the metric projectio Proj C of H oto C has the followig basic properties: i Proj C x Proj C y x y, for all x, y H; ii x y, Proj C x Proj C y Proj C x Proj C y 2, for every x, y H; iii x Proj C x,y Proj C x 0, for all x H, y C. Next we adopt the followig otatio: i x x meas that x coverges strogly to x; ii x xmeas that x coverges weakly to x; iii ω w x : {x : x j x} is the weak ω-limit set of the sequece {x }. Lemma 2.1 see 32 Demiclosedess Priciple. Let C be a closed ad covex subset of a Hilbert space H, ad let T : C C be a oexpasive mappig with Fix T /. If{x } is a sequece i C weakly covergig to x ad if { I T x } coverges strogly to y, the I T x y. 2.3 I particular, if y 0, thex Fix T.

5 Abstract ad Applied Aalysis 5 Lemma 2.2 see 33. Let C be a closed covex subset of H. Let{x } be a sequece i H ad x 0 H.If{x } is such that ω w x C ad satisfies the coditio x x 0 x0 Proj C x 0, 0, 2.4 the x Proj C x 0. Lemma 2.3 see 34. Assume {a } is a sequece of oegative real umbers such that a 1 1 γ ) a δ, 2.5 where {γ } is a sequece i 0, 1 ad {δ } is a sequece such that 1 1 γ ; 2 lim sup δ /γ 0 or 1 δ <. The lim a 0. Lemma 2.4 see 35. Let {x } ad {y } be bouded sequeces i a Baach space X, ad let {β } be a sequece i 0, 1 with 0 < lim if β lim sup β < Suppose x 1 1 β ) y β x, 2.7 for all 0, ad lim sup y 1 y x 1 x ) The, lim y x Mai Results I this sectio, we will state ad prove our mai results. Theorem 3.1. Let C be a closed covex subset of a real Hilbert space H. Letf : C R be a realvalued Fréchet differetiable covex fuctio. Assume that the solutio set S of 1.1 is oempty. Assume that the gradiet f is L-Lipschitzia. Let h : C H be a ρ-cotractio with ρ 0, 1. Let {x } be a sequece geerated by the followig hybrid gradiet projectio algorithm: x 1 Proj C θ h x 1 θ x γ f x ), 0, 3.1

6 6 Abstract ad Applied Aalysis where the sequeces {θ } 0, 1 ad {γ } 0, satisfy the followig coditios: i lim θ 0, 0 θ ad 0 θ 1 θ < ; ii 0 < lim if γ lim sup γ < 2/L ad 0 γ 1 γ <. The the sequece {x } geerated by 3.1 coverges to a miimizer x of 1.1 which is the uique solutio of the followig variatioal iequality: x S, I h x, x x 0, x S. 3.2 Proof. Take ay x S. Sice x C solves the miimizatio problem 1.1 if ad oly if x solves the fixed-poit equatio, x Proj C I γ f x for ay fixed positive umber γ. So, we have x Proj C I γ f x for all 0. It ca be rewritte as x Proj C θ x 1 θ x γ )) f x, θ From coditio ii 0 < lim if γ lim sup γ < 2/L, there exist two costats a ad b such that 0 <a γ b<2/l for sufficietly large ; without loss of geerality, we ca assume 0 <a γ b<2/l for all. Sice lim θ 0, without loss of geerality, we ca assume that 0 < θ < 1 bl/2 for all 0. So, 0 < lim if γ / 1 θ lim sup γ / 1 θ < 2/L. Hece, I γ / 1 θ f is oexpasive. From 3.1, weget x 1 x ProjC θ h x 1 θ x γ f x ) Proj C I γ f ) x Proj C θ h x 1 θ x γ )) f x 1 θ Proj C θ x 1 θ x γ f x )) 1 θ θ h x x 1 θ I γ ) f x 1 θ I γ ) f x 1 θ 3.4 θ h x h x θ h x x 1 θ x x θ ρ x x θ h x x 1 θ x x 1 1 ρ ) ) θ x x θ h x x { } max x x 1, 1 ρ h x x. Thus, we deduce by iductio that { } x x max x 0 x 1, 1 ρ h x x. 3.5

7 Abstract ad Applied Aalysis 7 This idicates that the sequece {x } is bouded ad so are the sequeces {h x } ad { f x }. The, we ca chose a costat M>0 such that { sup h x x f x } M Next, we estimate x 1 x.by 3.1, we have x 1 x ProjC θ h x 1 θ x γ f x ) Proj C θ 1 h x 1 1 θ 1 x 1 γ 1 f x 1 ) 1 θ x γ f x ) 1 θ 1 x 1 γ 1 f x 1 ) θ h x θ 1 h x 1 1 θ I γ ) f x 1 θ I γ ) f x 1 1 θ 1 θ θ 1 θ x 1 ) γ 1 γ f x 1 θ h x h x 1 θ θ 1 h x 1 1 θ I γ ) f x I γ ) f x 1 θ θ 1 x 1 1 θ 1 θ 3.7 γ γ 1 f x 1 θ h x h x 1 θ θ 1 h x ρ ) ) θ x x 1 θ θ 1 h x 1 x 1 γ γ 1 f x ρ ) ) θ x x 1 γ γ 1 θ θ 1 ) M. The, we ca combie the last iequality ad Lemma 2.3 to coclude that lim x 1 x Now we show that the weak limit set ω w x S. Choose ay x ω w x. The there must exist a subsequece {x j } of {x } such that x j x. At the same time, the real umber sequece {γ j } is bouded. Thus, there exists a subsequece {γ ji } of {γ j } which coverges

8 8 Abstract ad Applied Aalysis to γ. Without loss of geerality, we may assume that γ j γ. Notethat0< lim if γ lim sup γ < 2/L. So,γ 0, 2/L. Thatis,γ j γ 0, 2/L as j. Next, we oly eed to show that x S. First,from 3.8 we have that x j 1 x j 0. The, we have ) x j Proj C I γ f xj ) x j x j 1 x j 1 Proj C I γ j f x j ) ) Proj C I γ j f x j Proj C I γ f xj ) ) )) Proj C θ j h x j 1 θ j x j γ j f x j ) Proj C I γ j f x j ) ) xj Proj C I γ j f x j Proj C I γ f xj x j 1 ) ) ) h xj xj θ j x j γ j γ f x j x j Sice γ 0, 2/L, Proj C I γ f is oexpasive. It the follows from Lemma 2.1 demiclosedess priciple that x Fix Proj C I γ f. Hece, x S because of S Fix Proj C I γ f.so,ω w x S. Fially, we prove that x x, where x is the uique solutio of the VI 3.2.First,we show that lim sup I h x, x x 0. Observe that there exists a subsequece {x j } of {x } satisfyig lim sup I h x, x x lim I h x, x j x. j 3.10 Sice {x j } is bouded, there exists a subsequece {x ji } of {x j } such that x ji loss of geerality, we assume that x j x. The, we obtai x. Without lim sup I h x, x x lim I h x, x j x I h x, x x 0. j 3.11 By usig the property ii of Proj C, we have x 1 x 2 Proj C θ h x 1 θ x γ f x ) Proj C θ x 1 θ x γ f x ) 2

9 Abstract ad Applied Aalysis 9 θ h x x 1 θ I γ ) f x I 1 θ θ h x h x,x 1 x θ h x x, x 1 x 1 θ I γ ) f x I γ ) f x 1 θ 1 θ x 1 x γ ) ) f x,x 1 x 1 θ θ ρ x x x 1 x θ h x x, x 1 x 1 θ x x x 1 x 1 1 ρ ) θ ) x x x 1 x θ h x x, x 1 x 1 1 ρ ) θ x x x 1 x 2 θ h x x, x 1 x It follows that x 1 x ρ ) θ ) x x 2 1 ρ ) { } 2 θ 1 ρ h x x, x 1 x From Lemma 2.3, 3.11,ad 3.13, we deduce that x x. This completes the proof. From Theorem 3.1, we obtai immediately the followig theorem. Theorem 3.2. Let C be a closed covex subset of a real Hilbert space H. Letf : C R be a real-valued Fréchet differetiable covex fuctio. Assume S /. Assume that the gradiet f is L- Lipschitzia. Let {x } be a sequece geerated by the followig hybrid gradiet projectio algorithm: x 1 Proj C 1 θ x γ f x ), 0, 3.14 where the sequeces {θ } 0, 1 ad {γ } 0, satisfy the followig coditios: i lim θ 0, 0 θ ad 0 θ 1 θ < ; ii 0 < lim if γ lim sup γ < 2/L ad 0 γ 1 γ <. The the sequece {x } geerated by 3.14 coverges to a miimizer x of 1.1 which is the miimum orm elemet i S. Proof. I Theorem 3.1, weotethath is a o-self mappig from C to the whole space H. Hece, if we chose h x 0 for all x C, the Algorithm 3.1 reduces to Ad sequece x coverges strogly to x Proj S 0 which is obviously the miimum orm elemet i S. The proof is completed. Next, we suggest aother simple algorithm for droppig the assumptio 0 θ 1 θ <.

10 10 Abstract ad Applied Aalysis Theorem 3.3. Let C be a closed covex subset of a real Hilbert space H. Letf : C R be a real-valued Fréchet differetiable covex fuctio. Assume S /. Assume that the gradiet f is L- Lipschitzia. Let {x } be a sequece geerated by the followig hybrid gradiet projectio algorithm: x 1 Proj C I γ f ) ProjC 1 θ x, 0, 3.15 where γ 0, 2/L is a costat ad the sequeces {θ } 0, 1 satisfy the followig coditios: 1 lim θ 0; 2 0 θ. The the sequece {x } geerated by 3.15 coverges to a miimizer x of 1.1 which is the miimum orm elemet i S. Proof. Claim 1. The sequece {x } is bouded. Take x S. The we have x 1 x Proj C I γ f ) ProjC 1 θ x Proj C I γ f ) x 1 θ x x 1 θ x x θ x 3.16 max{ x x, x }. By iductio, x x max{ x 0 x, x } Claim 2. x Proj C I γ f x 0adω w x S. By the similar argumet as that i 18, page 366, we ca write Proj C I γ f ) 1 β ) I βt, 3.18 where T is oexpasive ad β 2 γl /4 0, 1. The we ca rewrite 3.15 as x 1 [ 1 β ) I βt ] Proj C 1 θ x 1 β ) x βtproj C 1 θ x 1 β ) ) Proj C 1 θ x x β ) x βy, where y T Proj C 1 θ x 1 β ) ProjC 1 θ x x β

11 Abstract ad Applied Aalysis 11 It follows that y 1 y T Proj C 1 θ 1 x 1 1 β ) ProjC 1 θ 1 x 1 x 1 β T Proj C 1 θ x 1 β ) ProjC 1 θ x x β T ProjC 1 θ 1 x 1 T Proj C 1 θ x 1 β Proj β C 1 θ 1 x 1 x 1 1 β ProjC 1 θ x x β 3.21 x 1 x θ 1 β x 1 θ β x. So, This together with Lemma 2.4 implies that lim sup y 1 y x 1 x ) lim y x Thus, lim x 1 x lim 1 β ) y x Note that ) x Proj C I γ f x x x 1 x 1 Proj C I γ f ) x x x 1 ProjC I γ f ) ProjC 1 θ x Proj C I γ f ) x 3.25 x x 1 θ x. Therefore, ) lim x Proj C I γ f x Now repeatig the proof of Theorem 3.1, we coclude that ω w x S.

12 12 Abstract ad Applied Aalysis Claim 3. lim sup x, x x 0 where x is the miimum orm elemet i S. Observe that there exists a subsequece {x j } of {x } satisfyig lim sup x, x x lim x, x j x. j 3.27 Sice {x j } is bouded, there exists a subsequece {x ji } of {x j } such that x ji x S. Without loss of geerality, we assume that x j x S. The, we obtai lim sup x, x x lim x, x j x x, x x 0. j 3.28 Claim 4. x x. From 3.15, we have x 1 x 2 ProjC I γ f ) ProjC 1 θ x Proj C I γ f ) x 2 1 θ x x 2 1 θ x x θ x θ 2 x x 2 2θ 1 θ x, x x θ 2 x 2 1 θ x x 2 θ { 2 1 θ x, x x θ x 2}. It is obvious that lim sup 2 1 θ x, x x θ x 2 0. The we ca apply Lemma 2.3 to the last iequality to coclude that x x. The proof is completed. Next, we suggest aother algorithm with the additioal projectios applied to the gradiet projectio algorithm. We show that this algorithm has strog covergece. Theorem 3.4. Let C be a closed covex subset of a real Hilbert space H. Letf : C R be a real-valued Fréchet differetiable covex fuctio. Assume S /. Assume that the gradiet f is L-Lipschitzia. Let x 0 H. For C 1 C ad x 1 Proj C1 x 0, defie a sequece {x } of C as follows: y Proj C x γ f x ), 0, C { z C 1 : y z x z }, 3.30 x 1 Proj C x 0. where the sequece {γ } 0, satisfies the coditio 0 < lim if γ lim sup γ < 2/L. The the sequece {x } geerated by 3.30 coverges to x Proj S x 0.

13 Abstract ad Applied Aalysis 13 Proof. It is obvious that C is covex. For ay x S, we have y x ProjC x γ f x ) Proj C x γ f x ) x x This implies that x C. Hece, S C.Fromx 1 Proj C x 0, we have x 0 x 1,x 1 y 0, y C Sice S C, we have x 0 x 1,x 1 u 0, u S So, for u S, we have 0 x 0 x 1,x 1 u x 0 x 1,x 1 x 0 x 0 u x 0 x 1 2 x 0 x 1,x 0 u 3.34 x 0 x 1 2 x 0 x 1 x 0 u. Hece, x 0 x 1 x 0 u, u S This implies that {x } is bouded. From x Proj C 1 x 0 ad x 1 Proj C x 0 C C 1, we have x 0 x,x x Hece, 0 x 0 x,x x 1 x 0 x,x x 0 x 0 x 1 x 0 x 2 x 0 x,x 0 x x 0 x 2 x 0 x x 0 x 1, ad therefore x 0 x x 0 x

14 14 Abstract ad Applied Aalysis This implies that lim x x 0 exists From 3.36 ad 3.39, weobtai x 1 x 2 x 1 x 0 x x 0 2 x 1 x 0 2 x x x 1 x,x x 0 x 1 x 0 2 x x By the fact x 1 C,weget y x 1 x x Therefore, from 3.40 ad 3.41, we deduce lim x y lim x Proj C I γ f ) x Now 3.42 ad Lemma 2.1 guaratee that every weak limit poit of {x } is a fixed poit of Proj C I γ f. Thatis,ω w x Fix Proj C I γ f S. At the same time, if we choose u Proj S x 0 i 3.35, we have x 0 x 1 x 0 Proj S x This fact ad Lemma 2.2 esure the strog covergece of {x } to Proj S x 0. This completes the proof. Now we give some remarks o our variat gradiet projectio methods. Remark 3.5. Uder the same cotrol parameters, the gradiet projectio methods 3.1 ad 1.6 are all strog coverget. However, 3.1 seems to have more advatage tha 1.6 as h is a o-self-mappig. Remark 3.6. The gradiet projectio method 3.14 is similar to 1.5 by usig 1 θ x istead of x.but 3.1 has strog covergece, ad especially 3.1 coverges strogly to the miimum orm elemet of S. Remark 3.7. The advatage of the gradiet projectio method 3.15 is that it has strog covergece uder some weaker assumptios o parameter θ. Remark 3.8. The gradiet projectio method 3.30 is simpler tha 1.7.

15 Abstract ad Applied Aalysis 15 Ackowledgmets Y. Yao was supported i part by NSFC ad NSFC G0105. Y.-C. Liou was supported i part by NSC E C.-F. We was supported i part by NSC M Refereces 1 E. M. Gafi ad D. P. Bertsekas, Two-metric projectio methods for costraied optimizatio, SIAM Joural o Cotrol ad Optimizatio, vol. 22, o. 6, pp , P. H. Calamai ad J. J. Moré, Projected gradiet methods for liearly costraied problems, Mathematical Programmig, vol. 39, o. 1, pp , E. S. Leviti ad B. T. Polyak, Costraied miimizatio methods, USSR Computatioal Mathematics ad Mathematical Physics, vol. 6, o. 5, pp. 1 50, B. T. Polyak, Itroductio to Optimizatio, Optimizatio Software, New York, NY, USA, A. Ruszczyński, Noliear Optimizatio, Priceto Uiversity Press, Priceto, NJ, USA, C. Wag ad N. Xiu, Covergece of the gradiet projectio method for geeralized covex miimizatio, Computatioal Optimizatio ad Applicatios, vol. 16, o. 2, pp , N. Xiu, C. Wag, ad J. Zhag, Covergece properties of projectio ad cotractio methods for variatioal iequality problems, Applied Mathematics ad Optimizatio, vol. 43, o. 2, pp , N. Xiu, C. Wag, ad L. Kog, A ote o the gradiet projectio method with exact stepsize rule, Joural of Computatioal Mathematics, vol. 25, o. 2, pp , M. Su ad H. K. Xu, Remarks o the gradiet-projectio algorithm, Joural of Noliear Aalysis ad Optimizatio, vol. 1, pp , Y. Cesor ad T. Elfvig, A multiprojectio algorithm usig Bregma projectios i a product space, Numerical Algorithms, vol. 8, o. 2 4, pp , C. Byre, A uified treatmet of some iterative algorithms i sigal processig ad image recostructio, Iverse Problems, vol. 20, o. 1, pp , Y. Cesor, T. Elfvig, N. Kopf, ad T. Bortfeld, The multiple-sets split feasibility problem ad its applicatios for iverse problems, Iverse Problems, vol. 21, o. 6, pp , Y. Cesor, T. Bortfeld, B. Marti, ad A. Trofimov, A uified approach for iversio problems i itesity-modulated radiatio therapy, Physics i Medicie ad Biology, vol. 51, o. 10, pp , H.-K. Xu, A variable Krasosel skii Ma algorithm ad the multiple-set split feasibility problem, Iverse Problems, vol. 22, o. 6, pp , H.-K. Xu, Iterative methods for the split feasibility problem i ifiite-dimesioal Hilbert spaces, Iverse Problems, vol. 26, o. 10, Article ID , G. Lopez, V. Marti, ad H.-K. Xu, Perturbatio techiques for oexpasive mappigs with applicatios, Noliear Aalysis: Real World Applicatios, vol. 10, o. 4, pp , G. Lopez, V. Marti, ad H. K. Xu, Iterative algorithms for the multiple-sets split feasibility problem, i Biomedical Mathematics: Promisig Directios i Imagig, Therapy Plaig ad Iverse Problems, Y. Cesor, M. Jiag, ad G. Wag, Eds., pp , Medical Physics Publishig, Madiso, Wis, USA, H.-K. Xu, Averaged mappigs ad the gradiet-projectio algorithm, Joural of Optimizatio Theory ad Applicatios, vol. 150, o. 2, pp , M. V. Solodov ad B. F. Svaiter, A ew projectio method for variatioal iequality problems, SIAM Joural o Cotrol ad Optimizatio, vol. 37, o. 3, pp , K. Goebel ad W. A. Kirk, Topics i Metric Fixed Poit Theory, vol. 28 of Cambridge Studies i Advaced Mathematics, Cambridge Uiversity Press, Cambridge, UK, C. Martiez-Yaes ad H.-K. Xu, Strog covergece of the CQ method for fixed poit iteratio processes, Noliear Aalysis: Theory, Methods & Applicatios, vol. 64, o. 11, pp , H.-K. Xu, Iterative algorithms for oliear operators, Joural of the Lodo Mathematical Society, vol. 66, o. 1, pp , T. Suzuki, Strog covergece theorems for ifiite families of oexpasive mappigs i geeral Baach spaces, Fixed Poit Theory ad Applicatios, vol. 2005, o. 1, pp , 2005.

16 16 Abstract ad Applied Aalysis 24 S. Reich ad H.-K. Xu, A iterative approach to a costraied least squares problem, Abstract ad Applied Aalysis, o. 8, pp , A. Sabharwal ad L. C. Potter, Covexly costraied liear iverse problems: iterative least-squares ad regularizatio, IEEE Trasactios o Sigal Processig, vol. 46, o. 9, pp , H. K. Xu, A iterative approach to quadratic optimizatio, Joural of Optimizatio Theory ad Applicatios, vol. 116, o. 3, pp , F. Ciaciaruso, G. Mario, L. Muglia, ad Y. Yao, A hybrid projectio algorithm for fidig solutios of mixed equilibrium problem ad variatioal iequality problem, Fixed Poit Theory ad Applicatios, vol. 2010, Article ID , 19 pages, F. Ciaciaruso, G. Mario, L. Muglia, ad Y. Yao, O a two-step algorithm for hierarchical fixed poit problems ad variatioal iequalities, Joural of Iequalities ad Applicatios, vol. 2009, Article ID , 13 pages, Y. Yao, Y. J. Cho, ad Y.-C. Liou, Algorithms of commo solutios for variatioal iclusios, mixed equilibrium problems ad fixed poit problems, Europea Joural of Operatioal Research, vol. 212, o. 2, pp , Y. Yao, Y.-C. Liou, ad S. M. Kag, Two-step projectio methods for a system of variatioal iequality problems i Baach spaces, Joural of Global Optimizatio. I press. 31 Y. Yao, R. Che, ad Y.-C. Liou, A uified implicit algorithm for solvig the triple-hierarchical costraied optimizatio problem, Mathematical Mathematical & Computer Modellig, vol. 55, pp , H. H. Bauschke ad J. M. Borwei, O projectio algorithms for solvig covex feasibility problems, SIAM Review, vol. 38, o. 3, pp , K. C. Kiwiel ad B. Łopuch, Surrogate projectio methods for fidig fixed poits of firmly oexpasive mappigs, SIAM Joural o Optimizatio, vol. 7, o. 4, pp , K. C. Kiwiel, The efficiecy of subgradiet projectio methods for covex optimizatio. I. Geeral level methods, SIAM Joural o Cotrol ad Optimizatio, vol. 34, o. 2, pp , K. C. Kiwiel, The efficiecy of subgradiet projectio methods for covex optimizatio. II. Implemetatios ad extesios, SIAM Joural o Cotrol ad Optimizatio, vol. 34, o. 2, pp , 1996.

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