Hybrid CQ projection algorithm with line-search process for the split feasibility problem

Transcription

1 Dang et al. Journal of Inequalities and Applications :106 DOI /s R E S E A R C H Open Access Hbrid CQ proection algorithm with line-search process for the split feasibilit problem Yazheng Dang 1*, Zhonghui Xue and Bo Wang 1 * Correspondence: gdz@163.com 1 School of Management, Universit of Shanghai for Science and Technolog, No. 516 Jungong Road, Shanghai, 00093, China ull list of author information is available at the end of the article Abstract In this paper, we propose a hbrid CQ proection algorithm with two proection steps and one Armio-tpe line-search step for the split feasibilit problem. The line-search technique is intended to construct a hperplane that strictl separates the current point from the solution set. The net iteration is obtained b the proection of the initial point on a regress region the intersection of three sets. Hence, algorithm converges faster than some other algorithms. Under some mild conditions, we show the convergence. Preliminar numerical eperiments show that our algorithm is efficient. MSC: 47H05; 47J05; 47J5 Kewords: split feasible problem; Armio-tpe line-search technique; proection algorithm; convergence 1 Introduction Split feasibilit problem SP is to find a point satisfing C, A Q, 1.1 where C and Q are nonempt conve sets in R N and R M,respectivel,andA is an M b N real matri. The SP was originall introduced in [1] and has broad applications in man fields, such as image reconstruction problem, signal processing, and radiation therap [ 4]. Various algorithms have been invented to solve it see [5 16] and references therein. The well-nown CQ algorithm presented in [1] is defined as follows: Denote b P C the orthogonal proection onto C, thatis,p C =arg min C for C; thentaean initial point 0 arbitraril and define the iterative step b +1 = P C I γ A T I P Q A, 1. where 0 < γ </ρa T A, and ρa T A is the spectral radius of A T A. The algorithms mentioned use a fied stepsize restricted b the Lipschitz constant L, which depends on the largest eigenvalue spectral radius of the matri. We now that 016 Dang et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in an medium, provided ou give appropriate credit to the original authors and the source, provide a lin to the Creative Commons license, and indicate if changes were made.

2 Dang et al. Journal of Inequalities and Applications :106 Page of 11 computing the largest eigenvalue ma be ver hard and conservative estimate of the constant L usuall results in slow convergence. To overcome the difficult in estimating the Lipschitz constant, He et al. [17] developed a selfadaptive method for solving variational problems.the numerical resultsreportedin [17] have shown that the selfadaptive strateg is valid and robust for solving variational inequalit problems. Subsequentl, man selfadaptive proection methods were presented to solve the split feasibilit problem [18, 19]. On the other hand, hbrid proection method was developed b Naao and Taahashi [0], Kamimure and Taahashi [1], and Martines-Yanes and Xu [] tosolvetheprob- lem of finding a common element of the set of fied points of a nonepansive mapping and the set of solutions of an equilibrium problem. Man modified hbrid proection methods were presented to solve different problems [3, 4]. In this paper, motivated b the selfadaptive method and hbrid proection method for solving variational inequalit problem, based on the CQ algorithm for the SP, we propose a hbrid CQ proection algorithm for the split feasibilit problem, which uses different variable stepsizes in two proection steps. Algorithm performs a computationall inepensive Armio-tpe linear search along the search direction in order to generate separating hperplane, which is different from the general selfadaptive Armio-tpe procedure [18, 19]. or the second proection step, we select the proection onto the intersection of the set C with the halfspaces, which maes an optimal stepsize available at each iteration and hence guarantees that the net iteration is the closest to the solution set. Therefore, the iterative sequence generated b the algorithm converges more quicl. The algorithm is shown to be convergent to a point in the solution set under some assumptions. The paper is organized as follows. In Section, we recall some preliminaries. In Section 3, we propose a hbrid CQ proection algorithm for the split feasibilit problem and show its convergence. In Section 4, we give an eample to test the efficienc. In Section 5, we give some concluding remars. Preliminaries We denote b I the identit operator and b itthesetoffiedpointsofanoperatort, that is, it:={ = T}. Recall that a mapping T : R n Ris said to be monotone if T T, 0,, R n. or a monotone mapping T,ifT T, =0iff =,thenitissaidtobestrictl monotone. A mapping T : R n R n is called nonepansive if T T,, R n. Lemma.1 Let be a nonempt closed and conve subset in H. Then, for an, H and z, it is well nown that the following statements hold: 1 P, z P 0. P P, 0.

3 Dang et al. Journal of Inequalities and Applications :106 Page 3 of 11 3 P P,, R n, or more precisel, P P P + P. 4 P z z P. Remar.1 In fact, the proection propert 1 also provides a sufficient and necessar condition for a vector u K to be the proection of the vector ; thatis,u = P K ifand onl if u, z u 0, z K. Throughout the paper, we b denote Ɣ the solution set of split feasibilit problem, that is, Ɣ := { C A Q}..1 3 Algorithm and convergence analsis Let := A T I P Q A. rom [10]wenowthat is Lipschitz-continuous with constant L =1/ρA T Aandis 1 A - inverse strongl monotone. We first note that the solution set coincides with zeros of the following proected residual function: e:= P C, e, μ:= PC μ ; with this definition, we have e,1=e, and Ɣ if and onl if e, μ=0.oran R N and α 0, define α=p C α, e, α= α. The following lemma is useful for the convergence analsis in the net section. Lemma 3.1 [18] Let be a mapping from R N into R N. or an R N and α 0, we have min{1, α} e,1 e, α ma{1, α} e,1. The detailed iterative processes are as follows: Algorithm 3.1 Step 0. Choose an arbitrar initial point 0 C, η 0 > 0, and three parameters γ 0, 1, 0, 1, and θ >1,andset =0. Step 1. Given the current iterative point,compute z = P C [ ], 3.1 where := min{θη 1,1}.Obviousl,e, = z.ife, =0,thenstop;

4 Dang et al. Journal of Inequalities and Applications :106 Page 4 of 11 Step. Compute = η e,, where η = γ m with m being the smallest nonnegative integer m satisfing γ e,, e, e,. 3. Step 3. Compute +1 = P C H 1 H 0, 3.3 where H 1 = { R N, 0 }, H = { R N, 0 0 }. Select = +1andgotoStep1. Remar 3.1 1Inthealgorithm,aproectionfromR N onto the intersection C H 1 H needs to be computed, that is, procedure 3.3 at each iteration. Surel, if the domain set C has a special structure such as a bo or a ball, then the net iteration +1 can easil be computed. If the domain set C is defined b a set of linear inequalities, then computing the proection is equivalent to solving a strictl conve quadratic optimization problem. It can readil be verified that the hperplane H 1 strictl separates the current point from the solution set Ɣ if is not a solution of the problem. That is, Ɣ H,andthe hperplane H strictl separates the initial point 0 from the solution set Ɣ. 3 Compared with the general hbrid proection method in [1, ], besides the maor modification made in the proection domain in the last proection step, the values of some parameters involved in the algorithm are also adusted. Before establishing the global convergence of Algorithm 3.1,wefirstgivesomelemmas. Lemma 3. or all 0, thereeistsanonnegativenumber m satisfing 3.. Proof Suppose that, for some,3. is not satisfied for an integer m,that is, γ m e,, e, e,. 3.4 B the definition of e, and Lemma.1 we now that PC, P C 0. Then, e, 1 e, >0. 3.5

5 Dang et al. Journal of Inequalities and Applications :106 Page 5 of 11 Since γ 0, 1 and := min{θη 1,1},from3.4weget Hence, lim γ m e, =. m, e, e,. 3.6 But 3.6 contradicts 3.5 because <1and e, 0. Hence, 3. issatisfiedfor some integer m. The following lemma shows that the halfspace H 1 in Algorithm 3.1 strictl separates from the solution set Ɣ if Ɣ is nonempt. Lemma 3.3 If Ɣ, then the halfspace H 1 in Algorithm 3.1 separates the point from the set Ɣ. Moreover, Ɣ H 1 C, 0. Proof B the definition of e, and Algorithm 3.1 we have =1 η + η z = η e,, which can be rewritten as η e, =. Then, b this and b 3.weget, > Hence, b the definition of H 1 and b 3.7weget / H 1. On the other hand, for an Ɣ and C, b the monotonicit of we have, B the conveit of C it is eas to see that C. Letting = in 3.9, we have, 0, which implies H 1.Moreover,itiseastoseethatƔ H1 C, 0. The following lemma sas that if the solution set is nonempt, then Ɣ H 1 H C and thus H 1 H C is a nonempt set. Lemma 3.4 If the solution set Ɣ, then Ɣ H 1 H Cforall 0.

6 Dang et al. Journal of Inequalities and Applications :106 Page 6 of 11 Proof rom the previous analsis it is sufficient to prove that Ɣ H proof will be given b induction. Obviousl, if =0, then for all 0. The Ɣ H 0 = RN. Now, suppose that Ɣ H for = l 0. Then Ɣ H 1 l H l C. or an Ɣ, b Lemma.1 and the fact that l+1 = P H 1 l H l C 0 we have that l+1, 0 l+1 0. Thus, Ɣ Hl+1. This shows that Ɣ H for all 0, and the desired result follows. or the case that the solution set is empt, we have that Hl 1 Hl C is also nonempt from the following lemma, which implies the feasibilit of Algorithm 3.1. Lemma 3.5 Suppose that Ɣ =. Then Hl 1 Hl C for all 0. We net prove our main convergence result. Theorem 3.1 Suppose the solution set Ɣ is nonempt. Then the sequence { } generated b Algorithm 3.1 is bounded, and all its cluster points belong to the solution set. Moreover, the sequence { } globall converges to a solution such that = P Ɣ 0. Proof Tae an arbitrar point Ɣ.Then H 1 H C.Since +1 = P H 1 H C 0, bthedefinitionoftheproectionwehavethat So, { } is a bounded sequence, and so is { }. Since +1 H, from the definition of the proection operator it is obvious that P H +1 = +1.

7 Dang et al. Journal of Inequalities and Applications :106 Page 7 of 11 or, b the definition of H, for all z H,wehave z, 0 0. Obviousl, = P H 0. Thus, using Lemma.1,wehave P H +1 P H P H P H 0, that is, , which can be written as Thus, the sequence { 0 } is nondecreasing and bounded and hence convergent, which implies that lim +1 = On the other hand, b +1 H 1 we get +1, Since =1 η + η z = η e,, b 3.10wehave +1, = +1 + η e,, 0, which implies η e,, +1, 0. Using the Cauch-Schwarz inequalit and 3., we obtain η e, η e,, B Lemma 3.1, e, min{1, μ } e.

8 Dang et al. Journal of Inequalities and Applications :106 Page 8 of 11 Since = min{θη 1,1},wehave 1. Hence, e, min{1, } e e. Therefore, η e η e, +1. Taing into account that η = γ m,wefurtherobtain η e η e, Since is continuous, there eists a constant M such that M.B3.9and3.1 it follows that lim η e = or an convergent subsequence { } of { },itslimitisdenotedb,thatis, lim =. Now, we consider the two possible cases for Suppose first that {η } has a limit. Then lim η =0. B the choice of η in Algorithm 3.1 we now that η γ e,, e, e, Since lim η γ e, =. urthermore, lim η γ e, =. So, b 3.14weobtain lim η γ e,, e, =, e, lim e, = e, μ. 3.15

9 Dang et al. Journal of Inequalities and Applications :106 Page 9 of 11 Using the similar arguments of 3., we have, e, μ 1 e,. Since e, min{1, μ } e, from 3.15wegetthate =0,andthus is a solution of problem 1.1. Suppose now that lim sup η > 0. Because of 3.13, it must be that lim inf e =0.Since e is continuous, we get e =0,andthus is a solution of problem 1.1. Now, we prove that the sequence { } converges to a point contained in Ɣ. Let = P Ɣ 0. Since Ɣ, b Lemma 3.4 we have H 1 1 H 1 C for all. So, b the iterative sequence of Algorithm 3.1 we have 0 0. Thus, = = , , 0. Letting,wehave 0 + 0, 0 =, 0 0, where the last inequalit is due to Lemma.1 and the fact that = P Ɣ 0 and Ɣ.So, = = P Ɣ 0. Thus, the sequence { } has a unique cluster point P Ɣ 0, which shows the global convergence of { }. 4 Numerical eperiments To test the effectiveness of our algorithm in this paper, we implemented it in MATLAB to solve the following eample. We use e, ε =10 5 as the stopping criterion. We denote Algorithm 3.1 in [14]asAlgorithm3.1. Throughout the computational eperiment, the parameters used in Algorithm 3.1 were set as γ =0.6, =0.8,θ =1.5,η 0 =0.3, β = 1. The numerical results of the eample are given in Table 1, Iter denotes the number of iterations, CPU denotes the computing time, and denotes the approimate solution.

10 Dang et al. Journal of Inequalities and Applications :106 Page 10 of 11 Table 1 Results for Eample Initiative point Algorithm 3.1 with β =1 Algorithm = 3,, 0 Iter = 78; CPU s = Iter = 43; CPU s = = 0.038, , = 1.051;.3679; =0, 4,0 Iter = 39; CPU s = Iter = 0; CPU s = = , , =.0711, 1.634,.1036 Eample Let C = { R } and Q = { R3 3 1}, A = I. ind C with A Q. rom the numerical eperiments for the simple eample we can see that our proposed method has good convergence properties. 5 Some concluding remars This paper presented a hbrid CQ proection algorithm with two proection steps and one Armio linear-search step for solving the split feasibilit problem SP. Different from the self-adaptive proection methods proposed b Zhang et al. [18], we use a new liner-search rule, which ensures that the hperplane H separates the current from the solution set Ɣ. The net iteration is generated b the proection of the starting point on a shrining proection region the intersection of three sets. Preliminar numerical eperiments demonstrate a good behavior. However, whether the idea can be used to solve multiple-set SP deserves further research. Competing interests The authors declare that the have no competing interests. Authors contributions All authors contributed equall to the writing of this paper. All authors read and approved the final manuscript. Author details 1 School of Management, Universit of Shanghai for Science and Technolog, No. 516 Jungong Road, Shanghai, 00093, China. Henan Poltechnic Universit, Shii Road, Henan, , China. Acnowledgements This wor was supported b Natural Science oundation of Shanghai 14ZR14900 and Innovation Program of Shanghai Municipal Education Commission 15ZZ074. Received: 8 August 015 Accepted: 15 March 016 References 1. Censor, Y, Elfving, T: A multiproection algorithm using Bregman proections in a product space. Numer. Algorithms 8, Bne, C: An unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 0, Censor, Y, Motova, A, Segal, A: Perturbed proections and subgradient proections for the multiple-sets split feasibilit problem. J. Math. Anal. Appl. 37, Censor, T, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibilit problem and its applications for inverse problems. Inverse Probl. 1, Dang, Y, Gao, Y: Bi-etrapolated subgradient proection algorithm for solving multiple-sets split feasibilit problem. Appl. Math. J. Chin. Univ. Ser. B 93, Qu, B, Xiu, N: A new halfspace-relaation proection method for the split feasibilit problem. Linear Algebra Appl. 48, Dang, Y, Xue, ZH: Iterative process for solving a multiple-set split feasibilit problem. J. Inequal. Appl doi: /s Brne, C: Iterative oblique proection onto conve sets and the split feasibilit problem. Inverse Probl. 18, Xu, H: A variable Krasnosel si-mann algorithm and the multiple-set split feasibilit problem. Inverse Probl.,

11 Dang et al. Journal of Inequalities and Applications :106 Page 11 of Dang, Y, Gao, Y: The strong convergence of a KM-CQ-lie algorithm for split feasibilit problem. Inverse Probl doi: / /7/1/ Yan, AL, Wang, GY, Xiu, NH: Robust solutions of split feasibilit problem with uncertain linear operator. J. Ind. Manag. Optim. 3, Yang, Q: The relaed CQ algorithm solving the split feasibilit problem. Inverse Probl. 0, Zhao, J, Yang, Q: Several solution methods for the split feasibilit problem. Inverse Probl. 1, Qu, B, Xiu, NH: A note on the CQ algorithm for the split feasibilit problem. Inverse Probl. 1, Ansari, QH, Rehan, A: Split feasibilit and fied point problems. In: Nonlinear Analsis: Approimation Theor, Optimization and Applications, pp Springer, New Yor Latif, A, Sahu, DR, Ansari, QH: Variable KM-lie algorithms for fied point problems and split feasibilit problems. ied Point Theor Appl. 014, ArticleID He, B, Yang, H, Meng, Q, Han, D: Modified Goldstein-Levitin-Pola proection method for asmmetric strong monotone variational inequalities. J. Optim. Theor Appl. 11, Zhang, WX, Han, D, Li, ZB: A self-adaptive proection method for solving the multiple-sets split feasibilit problem. Inverse Probl. 5, Zhao, JL, Yang, Q: Self-adaptive proection methods for the multiple-sets split feasibilit problem. Inverse Probl. 7, Naao, K, Taahashi, W: Strong convergence theorems for nonepansive mappings and nonepansive semigroups. J. Math. Anal. Appl. 79, Kamimura, S, Taahashi, W: Strong convergence of a proimal-tpe algorithm in a Banach space. SIAM J. Optim. 133, Martinez-Yanes, C, Xu, H-K: Strong convergence of the CQ method for fied point iteration processes. Nonlinear Anal., Theor Methods Appl. 6411, Taahashi, W, Taeuchi, Y, Kubota, R: Strong convergence theorems b hbrid methods for families of nonepansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, Poom, K: A hbrid approimation method for equilibrium and fied point problems for a monotone mapping and a nonepansive mapping. Nonlinear Anal. Hbrid Sst.,