Hybrid CQ projection algorithm with line-search process for the split feasibility problem
- Yazheng Dang^{1}Email author,
- Zhonghui Xue^{2} and
- Bo Wang^{1}
https://doi.org/10.1186/s13660-016-1039-7
© Dang et al. 2016
Received: 8 August 2015
Accepted: 15 March 2016
Published: 31 March 2016
Abstract
In this paper, we propose a hybrid CQ projection algorithm with two projection steps and one Armijo-type line-search step for the split feasibility problem. The line-search technique is intended to construct a hyperplane that strictly separates the current point from the solution set. The next iteration is obtained by the projection 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. Preliminary numerical experiments show that our algorithm is efficient.
Keywords
split feasible problem Armijo-type line-search technique projection algorithm convergenceMSC
47H05 47J05 47J251 Introduction
The algorithms mentioned use a fixed stepsize restricted by the Lipschitz constant L, which depends on the largest eigenvalue (spectral radius) of the matrix. We know that computing the largest eigenvalue may be very hard and conservative estimate of the constant L usually results in slow convergence. To overcome the difficulty in estimating the Lipschitz constant, He et al. [17] developed a selfadaptive method for solving variational problems. The numerical results reported in [17] have shown that the selfadaptive strategy is valid and robust for solving variational inequality problems. Subsequently, many selfadaptive projection methods were presented to solve the split feasibility problem [18, 19]. On the other hand, hybrid projection method was developed by Nakajo and Takahashi [20], Kamimure and Takahashi [21], and Martines-Yanes and Xu [22] to solve the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem. Many modified hybrid projection methods were presented to solve different problems [23, 24].
In this paper, motivated by the selfadaptive method and hybrid projection method for solving variational inequality problem, based on the CQ algorithm for the SFP, we propose a hybrid CQ projection algorithm for the split feasibility problem, which uses different variable stepsizes in two projection steps. Algorithm performs a computationally inexpensive Armijo-type linear search along the search direction in order to generate separating hyperplane, which is different from the general selfadaptive Armijo-type procedure [18, 19]. For the second projection step, we select the projection onto the intersection of the set C with the halfspaces, which makes an optimal stepsize available at each iteration and hence guarantees that the next iteration is the ‘closest’ to the solution set. Therefore, the iterative sequence generated by the algorithm converges more quickly. 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 2, we recall some preliminaries. In Section 3, we propose a hybrid CQ projection algorithm for the split feasibility problem and show its convergence. In Section 4, we give an example to test the efficiency. In Section 5, we give some concluding remarks.
2 Preliminaries
We denote by I the identity operator and by \(\operatorname {Fix}(T)\) the set of fixed points of an operator T, that is, \(\operatorname {Fix}(T):=\{x | x=Tx\}\).
Lemma 2.1
- (1)
\(\langle P_{\Omega}(x)-x, z-P_{\Omega}(x)\rangle\geq0\).
- (2)
\(\langle P_{\Omega}(x)-P_{\Omega}(y), x-y\rangle\geq0\).
- (3)\(\Vert P_{\Omega}(x)-P_{\Omega}(y)\Vert \leq \Vert x-y\Vert \), \(\forall x,y \in \Re^{n}\), or more precisely,$$\bigl\Vert P_{\Omega}(x)-P_{\Omega}(y)\bigr\Vert ^{2} \leq \Vert x-y\Vert ^{2}-\bigl\Vert P_{\Omega}(x)-x+y-P_{\Omega}(y) \bigr\Vert ^{2}. $$
- (4)
\(\Vert P_{\Omega}(x)-z\Vert \leq \Vert x-z\Vert -\Vert P_{\Omega}(x)-x\Vert \).
Remark 2.1
3 Algorithm and convergence analysis
Lemma 3.1
[18]
The detailed iterative processes are as follows:
Algorithm 3.1
Step 0. Choose an arbitrary initial point \(x^{0}\in C\), \(\eta_{0}>0\), and three parameters \(\gamma\in(0,1)\), \(\sigma\in(0,1)\), and \(\theta >1\), and set \(k=0\).
Remark 3.1
(1) In the algorithm, a projection from \(\Re^{N}\) onto the intersection \(C\cap H_{k}^{1}\cap H_{k}^{2}\) needs to be computed, that is, procedure (3.3) at each iteration. Surely, if the domain set C has a special structure such as a box or a ball, then the next iteration \(x^{k+1}\) can easily be computed. If the domain set C is defined by a set of linear (in)equalities, then computing the projection is equivalent to solving a strictly convex quadratic optimization problem.
(2) It can readily be verified that the hyperplane \(H_{k}^{1}\) strictly separates the current point \(x^{k}\) from the solution set Γ if \(x^{k}\) is not a solution of the problem. That is, \(\Gamma\subset H_{k}^{-}\), and the hyperplane \(H_{k}^{2}\) strictly separates the initial point \(x^{0}\) from the solution set Γ.
(3) Compared with the general hybrid projection method in [21, 22], besides the major modification made in the projection domain in the last projection step, the values of some parameters involved in the algorithm are also adjusted.
Before establishing the global convergence of Algorithm 3.1, we first give some lemmas.
Lemma 3.2
For all \(k\geq0\), there exists a nonnegative number m satisfying (3.2).
Proof
The following lemma shows that the halfspace \(H_{k}^{1}\) in Algorithm 3.1 strictly separates \(x^{k}\) from the solution set Γ if Γ is nonempty.
Lemma 3.3
Proof
The following lemma says that if the solution set is nonempty, then \(\Gamma\subset H_{k}^{1}\cap H_{k}^{2}\cap C\) and thus \(H_{k}^{1}\cap H_{k}^{2}\cap C\) is a nonempty set.
Lemma 3.4
If the solution set \(\Gamma\neq\emptyset\), then \(\Gamma\subset H_{k}^{1}\cap H_{k}^{2}\cap C\) for all \(k\geq 0\).
Proof
For the case that the solution set is empty, we have that \(H_{l}^{1}\cap H_{l}^{2}\cap C\) is also nonempty from the following lemma, which implies the feasibility of Algorithm 3.1. □
Lemma 3.5
Suppose that \(\Gamma=\emptyset\). Then \(H_{l}^{1}\cap H_{l}^{2}\cap C\neq\emptyset\) for all \(k\geq 0\).
We next prove our main convergence result.
Theorem 3.1
Suppose the solution set Γ is nonempty. Then the sequence \(\{x^{k}\}\) generated by Algorithm 3.1 is bounded, and all its cluster points belong to the solution set. Moreover, the sequence \(\{x^{k}\}\) globally converges to a solution \(x^{\ast}\) such that \(x^{\ast}=P_{\Gamma}(x^{0})\).
Proof
Now, we consider the two possible cases for (3.13).
Suppose now that \(\limsup_{k\rightarrow\infty}\eta_{k}>0\). Because of (3.13), it must be that \(\liminf_{k\rightarrow\infty} \Vert e(x^{k})\Vert =0\). Since \(e(\cdot)\) is continuous, we get \(e(\bar{x})=0\), and thus x̄ is a solution of problem (1.1).
Now, we prove that the sequence \(\{x^{k}\}\) converges to a point contained in Γ.
4 Numerical experiments
Results for Example
Initiative point | Algorithm 3.1 \(\boldsymbol{^{\ast}}\) with β = 1 | Algorithm 3.1 |
---|---|---|
\(x^{0}=(3,2,0)\) | Iter = 78; CPU (s)=0.113 | Iter = 43; CPU (s)=0.098 |
\(x^{\ast}=(0.0238,0.0581,-0.1203)\) | \(x^{\ast}=(1.0512;-2.3679; 1.0613)\) | |
\(x^{0}=(0,-4,0)\) | Iter = 39; CPU (s)=0.096 | Iter = 20; CPU (s)=0.053 |
\(x^{\ast}=(-0.1608,-0.5033,0.8663)\) | \(x^{\ast}=(-2.0711,1.2634,2.1036)\) |
Example
Let \(C = \{x \in\Re^{3} | x_{1}^{2}+x_{2}^{2} \leq 40\}\) and \(Q=\{x\in\Re^{3}| x_{2}-x_{3}^{2}\leq1\}\), \(A=I\). Find \(x\in C \) with \(Ax\in Q\).
From the numerical experiments for the simple example we can see that our proposed method has good convergence properties.
5 Some concluding remarks
This paper presented a hybrid CQ projection algorithm with two projection steps and one Armijo linear-search step for solving the split feasibility problem (SFP). Different from the self-adaptive projection methods proposed by Zhang et al. [18], we use a new liner-search rule, which ensures that the hyperplane \(H_{k}\) separates the current \(x^{k}\) from the solution set Γ. The next iteration is generated by the projection of the starting point on a shrinking projection region (the intersection of three sets). Preliminary numerical experiments demonstrate a good behavior. However, whether the idea can be used to solve multiple-set SFP deserves further research.
Declarations
Acknowledgements
This work was supported by Natural Science Foundation of Shanghai (14ZR1429200) and Innovation Program of Shanghai Municipal Education Commission (15ZZ074).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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