Research Article Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem

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1 Abstract and Applied Analysis Volume 01, Article ID , 11 pages doi: /01/ Research Article Self-Adaptive and Relaxed Self-Adaptive Proection Methods for Solving the Multiple-Set Split Feasibility Problem Ying Chen, 1 Yuansheng Guo, Yanrong Yu, and Rudong Chen 1 Textile Division, Tianin Polytechnic University, Tianin , China Department of Mathematics, Tianin Polytechnic University, Tianin , China Correspondence should be addressed to Rudong Chen, chenrd@tpu.edu.cn Received 10 October 01; Accepted 17 November 01 Academic Editor: Yongfu Su Copyright q 01 Ying Chen et al. 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. Given nonempty closed convex subsets C i R m, i 1,,...,tand nonempty closed convex subsets Q R n, 1,,...,r,inthen- andm-dimensional Euclidean spaces, respectively. The multipleset split feasibility problem MSSFP proposed by Censor is to find a vector x t i1 C i such that Ax r 1 Q,whereA is a given M N real matrix. It serves as a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator s range. MSSFP has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing. In this paper, for the MSSFP, we first propose a new self-adaptive proection method by adopting Armio-like searches, which dose not require estimating the Lipschitz constant and calculating the largest eigenvalue of the matrix A T A; besides, it makes a sufficient decrease of the obective function at each iteration. Then we introduce a relaxed self-adaptive proection method by using proections onto half-spaces instead of those onto convex sets. Obviously, the latter are easy to implement. Global convergence for both methods is proved under a suitable condition. 1. Introduction The multiple-sets split feasibility problem MSSFP requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It is formulated as follows: Find a x C : t C i i1 r such that Ax Q : Q, 1 1.1

2 Abstract and Applied Analysis where nonempty closed convex sets C i R n,i 1,,...,t,inthen-dimensional Euclidean space R n, and nonempty closed convex sets Q R m, 1,,...,r,inthem-dimensional Euclidean space R m. A is an m n real matrix. Specially, the problem with only a single set C in R n and a single set Q in R m was introduced by Censor and Elfving 1 and was called the split feasibility problem SFP. Such MSSFPs 1.1, proposedin, arise in signal processing, image reconstruction and so on. Various algorithms have been invented to solve MSSFP 1.1. See 5 and references therein. In, Censor and Elfving were handling the MSSFP 1.1, for both the consistent and the inconsistent cases, where they aim at minimizing the proximity function ) 1 t ) 1 r Px α i P Ci x x PQ β Ax Ax. i For convenience reasons, they consider an additional closed convex set Ω R n. Their algorithm for the MSSFP 1.1 involves orthogonal proection onto Ω R n,c i R n,i 1,,...,t,andQ R m, 1,,...,r, which were assumed to be easily calculated and has the following iterative step: x k1 P Ω x k γ t α i P Ci x k) x k) i1 r β i A T P Q Ax k) Ax k), where α i > 0, i 1,,...,t. β > 0, 1,,...,r. γ 0, /L, L 1 t i1 α i λ r 1 β is the Lipschitz constant of Px, which is the gradient of the proximity function Px defined by 1., andλ is the spectral radius of the matrix A T A. For any starting vector x 0 R n,the algorithm converges to a solution of the MSSFP 1.1, whenever MSSFP 1.1 has a solution. In the inconsistent case, it find a point closest to all sets. This algorithm uses a fixed stepsize related to the Lipschitz constant L, which sometimes computing it may be hard. On the other hand, even if we ow the Lipschitz constant L, the method with fixed stepsize may lead to slow speed of convergence. In 005, Qu and Xiu 6 modified the CQ algorithm 7 and relaxed CQ algorithm 8 by adopting Armio-like searches to solve the SFP, where the second algorithm used orthogonal proections onto half-spaces instead of proections onto the original convex sets, ust as Yang s relaxed CQ algorithm 8. This may reduce a lot of work for computing proections, since proections onto half-spaces can be directly calculated. Motivated by Qu and Xiu s idea, Zhao and Yang in 4 introduce a self-adaptive proection method by adopting Armio-like searches to solve the MSSFP 1.1 and propose a relaxed self-adaptive proection method by using orthogonal proections onto half-spaces instead of these proections onto the original convex sets, which is more practical. But the same as Algorithm 1.3, Zhao and Yang s algorithm involves an addition proection P Ω. Though the MSSFP 1.1 includes the SFP as a special case, the Zhao and Yang s algorithm does not reduce to Qu and Xiu s modifications of the CQ algorithm 6. In this paper, We first proposed a self-adaptive method by adopting Armio-like searches to solve the MSSFP 1.1 without an addition proection P Ω, then a relaxed selfadaptive proection method was introduced which only involves orthogonal proections onto

3 Abstract and Applied Analysis 3 half-spaces, so that the algorithm is implementable. We need not estimate the Lipschitz constant and make a sufficient decrease of the obection function at each iteration; besides, these proection algorithms can reduce to the modifications of the CQ algorithm 6 when the MSSFP 1.1 is reduced to the SFP. We also show convergence the algorithms under mild conditions.. Preliminaries In this section, we give some definitions and basic results that will be used in this paper. Definition.1. An operator F from a set X R n into R n is called a monotone on X, if Fx F y ),x y 0, x, y X;.1 b cocoercive on X with constant α>0, if Fx F y ),x y α Fx F y ), x, y X;. c Lipschitz continuous on X with constant λ>0, if Fx F y ) λ x y, x, y X..3 In particular, if λ 1, F is said to be nonexpansive. It is easily seen from the definitions that cocoercive mappings are monotone. Definition.. Functions fx, differentiable on a nonempty convex set S, is pseudoconvex if for every x 1,x S, the condition fx 1 <fx implies that fx T x 1 x 0..4 It is own that differentiable convex functions are pseudoconvex see 9. For a given nonempty closed convex set Ω in R n, the orthogonal proection from R n onto Ω is defined by P Ω x argmin { x y y Ω }, x R n..5 Lemma.3 see 10. Let Ω be a nonempty closed convex subset in R n, then, for any x, y R n and z Ω. 1 P Ω x x, z P Ω x 0; P Ω x P Ω y P Ω x P Ω y,x y ; 3 P Ω x z xz P Ω x x.

4 4 Abstract and Applied Analysis From Lemma.3 ), one sees that the orthogonal proection mapping P Ω x is cocoercive with modulus 1, monotone, and nonexpansive. Let F be a mapping from R n into R n. For any x R n and α>0, define xα P Ω x αfx, ex, α x xα. Lemma.4 see 6. Let F be a mapping from R n into R n. For any x R n and α>0, one has min{1,α} ex, 1 ex, α max{1,α} ex, 1. Lemma.5 see 5, 9. Suppose h : R n R is a convex function, then it is subdifferentiable everywhere and its subdifferentials are uniformly bounded on any bounded subset of R n. 3. Self-Adaptive Proection Iterative Scheme and Convergence Results It is easily seen that if the solution set of MSSFP 1.1 is nonempty, then the MSSFP 1.1 is equivalent to the minimization problem of q over all x C : t i1 C i. qx is defined by ) 1 r PQ qx β Ax Ax, where β > 0. Note that the gradient of qx is r qx β A T I P Q )Ax Consider the following constrained minimization problem: min { qx, x C }. 3.3 We say that a point x C is a stationary point of the problem 3.3 if it satisfies the condition qx,x x 0, x C. 3.4 This optimization problem is proposed by Xu 3 for solving the MSSFP 1.1; the q defined by 3. is L-Lipschitzian with L A r 1 β and q is 1/L-ism. Algorithm 3.1. Given constant β>0, σ 0, 1. Letx 0 be arbitrary. For k 0, 1,..., calculate x k1 P Ck1 x k τ k qxx k), 3.5 where C n C n mod N and mod function takes values in 1,,...,N, τ k βγ l k and l k is the smallest nonnegative integer l such that q P Ck1 x k βγ l qxx k)) q x k) σ q x k),x k P Ck1 x k βγ l qxx k). 3.6

5 Abstract and Applied Analysis 5 Algorithm 3.1 need not estimate the largest eigenvalue of the matrix A T A,andthe stepsize τ k is chosen so that the obective function qx has a sufficient decrease. It is in fact a special case of the standard gradient proection method with the Armio-like search rule for solving the constrained optimization problem: min { gx; x Ω }, 3.7 where Ω R n is a nonempty closed convex set, and the function gx is continuously differentiable on Ω, then the following convergence result ensures the convergence of Algorithm 3.1. Lemma 3. see 6. Let g C 1 Ω be pseudoconvex and xk be an infinite sequence generated by the gradient proection method with the Armio-like searches. Then, the following conclusions hold: 1 lim k gx k inf{gx : x Ω}; /Ω, which denotes the set of the optimal solutions to 3.7, if and only if there exists at least one limit point of {x k }. In this case, {x k } converges to a solution of 3.7. Since the function qx is convex and continuously differentiable on C, therefore it is pseudoconvex. Then, by Lemma 3., one immediately obtains the following convergence result. Theorem 3.3. Let {x k } be a sequence generated by Algorithm 3.1, then the following conclusions hold: 1 {x k } is bounded if and only if the solution set of 3.3 is nonempty. In such a case, {x k } must converge to a solution of 3.3. {x k } is bounded and lim k qx k 0if and only if the MSSFP 1.1 is solvable. In such a case, {x k } must converge to a solution of MSSFP 1.1. However, in Algorithm 3.1, it costs a large amount of work to compute the orthogonal proections P Ci and P Q ; therefore, the same as Censor s method, these proections are assumed to be easily calculated. But, in some cases it is difficult or costs too much work to exactly compute the orthogonal proection, then the efficiency of these methods will be deeply affected. In what follows, one assume that the proections are not easily calculated. One present a relaxed self-adaptive proection method. Carefully speaking, the convex sets C i and Q satisfy the following assumptions. 1 The sets C i, i 1,,...,t. are given by C i {x R n c i x 0}, 3.8 where c i : R n R, i 1,,...,t, are convex functions. The sets Q, 1,,...,r. are given by Q { y R m q y ) 0 }, 3.9 where q : R m R, 1,,...,r, are convex functions.

6 6 Abstract and Applied Analysis For any x R n, at least one subgradient ξ i c i x can be calculated, where c i x is a generalized gradient subdifferential of c i x at x and it is defined as follows: c i x {ξ i R n c i z c i x ξ i,z x z R n } For any y R m, at least one subgradient η i q y can be calculated, where q y is a generalized gradient subdifferential of q y at y and it is defined as the following: q y ) { η R m q u q y ) η,u y u R m} In the kth iteration, let C k i {x R n c i x k) } ξ k i,x xk 0, 3.1 where ξ k i is an element in c i x k,i 1,,...,t. Consider Q k {y R m q Ax k) } η k,y Axk 0, 3.13 where η k is an element in q Ax k,1,,...,r. By the definition of the subgradient, it is clear that C i C k i,q Q k and the orthogonal proections onto C k i and Q k can be calculated 4, 6, 8. Define q k x 1 ) r 1 β PQ k Ax Ax, 3.14 where β > 0. Then q k x r 1 β A T I P Q k ) Ax For the Lipschiitz constant and the cocoercive modulus of q defined by 3. are not related to the nonempty closed convex sets C i and Q 3, one can obtain that the q k x is L-Lipschitzian with L A r 1 β and 1/L-ism. So, q k x is monotone. Algorithm 3.4. Given constant γ>0, α 0, 1 μ 0, 1.Letx 0 be arbitrary. For k 0, 1,,..., compute x k P C k x k ρ k q k x k)), k1 3.16

7 Abstract and Applied Analysis 7 where ρ k γα l k and l k is the smallest non-negative interger l such that q k x k) q k x k) x k x k μ ρ k Set x k1 P C k x k ρ k q k x k)). k Lemma 3.5. The Armio-like search rule 3.17 is well defined, and μα/l ρ k γ. Proof. Obviously, from 3.17, ρ k γ for k 0, 1,... We ow that ρ k /α must violate inequality 3.17.Thatis, q k x k) q k P C k x k ρ k k1 α q k x k))) μ x k P C k k1 x k ρ k /α ) q )) k x k ρ k /α Since q k is Lipschitz continuous with constant L, which together with 3.19, we have ρ k > μα L, 3.0 which completes the proof. Theorem 3.6. Let {x k } be a sequence generated by Algorithm 3.4. If the solution set of the MSSFP 1.1 is nonempty, the {x k } converges to a solution of the MSSFP 1.1. Proof. Let x be a solution of the MSSFP 1.1, then x P C x P Ci x,i1,,...,t.and Ax P Q Ax P Q x,1,,...,r. Since C i C k i,q Q k for all i and, we have x C k i, Ax Q k,and,q kx 0; thus, we have q k x 0for all k 0, 1,... Using the monotonicity of q k,wehaveforallk 0, 1,... q k x k) q k x, x k x This implies q k x k), x k x q k x, x k x Therefore, we have q k x k),x k1 x q k x k),x k1 x k. 3.3

8 8 Abstract and Applied Analysis Thus, using part 3 of Lemma.3 and 3.3, weobtain x k1 x PC k x k ρ k q k x k)) x k1 x k ρ k q k x k) x x k1 x k ρ k q k x k) x k x ρk q k x k),x k x ρ k q k x k),x k1 x k x k x ρk q k x k),x k1 x k x k x ρk q k x k),x k1 x k x k x k x k x k,x k1 x k x k1 x k x k1 x k x k x k x k1 x k x k x x k x k x k1 x k x k x k ρ k q k x k),x k1 x k. 3.4 Since x k P C k x k ρ k q k x k, x k1 C k k1 k1.bylemma.31, we have xk x k ρ k q k x k,x k1 x k 0; also, by search rule 3.17, it follows that x k1 x x k x x k x k x k1 x k x k x k ρ k q k x k),x k1 x k x k x k ρ k q k x k),x k1 x k x k x x k x k x k1 x k ρ k q k x k) q k x k),x k1 x k x k x x k x k x k1 x k ρ k q k x k) q k x k) x k1 x k x k x x k x k μ x k x k x k x 1 μ ) x k x k, 3.5

9 Abstract and Applied Analysis 9 which implies that the sequence { x k x } is monotonically decreasing and hence {x k } is bounded. Consequently, we get from 3.5 lim x k x k k On the other hand x k1 x k x k1 x k x k x k x k ρ k q k x k) x k ρ k q k x k) x k x k qk ρ k x k) q k x k) x k x k 3.7 μ 1 ) x k x k, which results in lim x k1 x k 0. k 3.8 Let x be an accumulation point of {x k } and x k n x, where {x k n } n1 is a subsequence of {x k }. We will show that x is a solution of the MSSFP 1.1. Thus, we need to show that x C t i1 C i and A x Q r 1 Q. Since x 1 C k n k n 1, for all n 1,,..., then by the definition of Ck n k n, we have 1 c ki 1 x k n ) ξ k n1 k n 1,xk n1 x k n Passing onto the limit in this inequality and taking into account 3.9 and Lemma 3., we obtain that c 1 x 0withk n. Because C 1 is repeated regularly of C 1,C,...,C t,so x C i for every 1 i t;thus, x C t i1 C i. Next, we need to show A x Q r 1 Q. Let e k x, ρ x P C k x ρ q k x,k 0, 1,,..., then e x k n,ρ x k n x k n and k1 we get from Lemmas.4 and 3.,and3.6 that lim e x k n, 1) lim k n k n x k n x k n min { } lim 1,ρ k n x k n x k n min { 1, ρ } 0, 3.30 where ρ μα/l. Since x is a solution point of MSSFP 1.1, sox C i,i 1,,...,t.By using Lemma.3, we have 0 ) )) )) x k n q x k n P C x k n q x k n,p 1 C x k n q x k n x 1 ) ) e x k n, 1 q x k n ),x k n x e x k n,

10 10 Abstract and Applied Analysis From Lemma.3, we see that orthogonal proection mappings are cocoercive with modulus 1, taking into account the fact that the mapping F is cocoercive with modulus 1 if and only if I F is cocoercive with modulus 1 6, weobtainfrom3.31 and q x 0 that ) ) x k n x,e x k n, 1 e x k n ), 1 q x k n ),e x k n, 1 ) q x k n q x,x k n x ) e x k n ), 1 q x k n ),e x k n, 1 r ) β A I T P Q Ax n) k A T I P Q )Ax,x k n x 1 ) e x k n ), 1 q x k n ),e x k n, 1 r 1 β I P Q ) Ax n) k I P Q ) e x k n ), 1 q x k n ),e x k n, 1 )Ax,Ax k n Ax r 1 β Ax k n P Q ) Ax k n. 3.3 Since q x k n q x k n q x L x k n x and {x k n } is abounded, the sequence q x k n is also bounded. Therefore, from 3.30 and 3.3, wegetforall 1,,...,that lim Axk n P Q Ax n) k k n Moreover, since P Q Ax k n Q k n, we have ) q Ax k n η k n,p Q ) Ax k n Ax k n 0, 1,...,r Again passing onto the limits and combining those inequalities with Lemma.5, we conclude by 3.33 that q A x 0, 1,...,r Thus, x C t i1 C i and A x Q r 1 Q.

11 Abstract and Applied Analysis 11 Therefore, x is a solution of the MSSFP 1.1. So we may use x in place of x in 3.3 and get that the sequence { x k x } is convergent. Furthermore, noting that a subsequence of {x k },thatis,{x k n }, converges to x, weobtainthatx k x, ask. This completes the proof. 4. Concluding Remarks This paper introduced two self-adaptive proection methods with the Armio-like searches for solving the multiple-set split feasibility problem MSSFP 1.1. They need not compute the additional proection P Ω and avoid the difficult task of estimating the Lipschitz constant. It makes a sufficient decrease of the obective function at each iteration; thus the efficiency is enhanced greatly. Moreover in the second algorithm, we use the relaxed proection technology to calculate orthogonal proections onto convex sets, which may reduce a large amount of computing work and make the method more practical. The corresponding convergence theories have been established. Acowledgment This paper was supported partly by NSFC Grants no References 1 Y. Censor and T. Elfving, A multiproection algorithm using Bregman proections in a product space, Numerical Algorithms, vol. 8, no., pp. 1 39, Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, vol. 1, no. 6, pp , H. K. Xu, A variable Krasnosel skii-mann algorithm and the multiple-set split feasibility problem, Inverse Problems, vol., no. 6, pp , J. L. Zhao and Q. Z. Yang, Self-adaptive proection methods for the multiple-sets split feasibility problem, Inverse Problems, vol. 7, no. 3, Article ID , G. Lopez, V. Martin, and H. K. Xu, Iterative algorithms for the multiple-sets split feasibility problem, in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, Eds., pp , Medical Physics, Madison, Wis, USA, B. Qu and N. H. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, vol. 1, no. 5, pp , C. Byrne, Iterative oblique proection onto convex sets and the split feasibility problem, Inverse Problems, vol. 18, no., pp , Q. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, vol. 0, no. 4, pp , R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, USA, E. H. Zarantonello, Proections on Convex Sets in Hilbert Space and Spectral Theory, Universityof Wisconsin Press, Madison, Wis, USA, 1971.

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