不同凸集条件下CQ算法的应用分析

针对CQ算法,通过定义不同条件的下非空闭凸集C和Q,并结合讨论稀疏角度的CT重建问题,在R^N空间中给出了5种不同的实现方案,每种实现方案相对于CT重建模型,具备不同的物理含义.给定相同的迭代步数,通过仿真试验,分别对不同方案的重建精度进行了分析,从而确定了在相同收敛条件下CQ算法在应用时的最佳方案,为分裂可行性问题及其扩展形式在工程领域的应用提供了新的思路.     

图像域增强约束卷积稀疏编码的稀疏角度CT重建算法研究

稀疏角度CT重建因其可以降低辐射剂量引起广泛关注,然而减少角度会降低重建图像质量,影响诊断结果分析.为解决上述问题,提出了图像域增强约束卷积稀疏编码的稀疏角度CT重建算法,该算法继承了卷积稀疏编码的优点,通过直接处理整幅图像提取特征,克服了字典学习因图像分块聚合引起的伪影.继而引入全变分正则项来增强图像域的约束,可以有效地进一步抑制噪声.通过几组稀疏角度的重建实验与不同算法对比,实验结果表明,所提算法在噪声抑制、伪影减少和图像细节恢复方面性能优越.     

多重集非凸分裂可行问题的非精确均值投影算法

在本文中,我们引入了非精确均值投影算法来求解多重集非凸分裂可行问题,其中这些非凸集合为半代数邻近正则集合.通过借助著名的Kurdyka-Lojasiewicz不等式理论,我们建立了算法的收敛性.     

强收敛的球松弛CQ算法及其应用

为了求解分裂可行问题,Yu等提出了一个球松弛CQ算法.由于该算法只需计算到闭球上的投影,同时不需要计算有界线性算子的范数,该算法是容易实现的.但是球松弛CQ算法在无穷维Hilbert空间中仅仅具有弱收敛性.首先构造了一个强收敛的球松弛CQ算法.在较弱的条件下,证明了算法的强收敛性.其次将该算法应用到一类闭凸集上的投影问...     

基于字典学习方法的CT不完全投影图像重建算法

对于不完全投影角度的重建研究是CT图像重建中一个重要的问题.将压缩感知中字典学习的方法与CT重建算法ART迭代算法相结合.字典学习方法中字典更新采用K-SVD(K-奇异值分解)算法,稀疏编码采用OMP(正交匹配追踪)算法.最后通过对标准Head头部模型进行仿真实验,验证了字典学习方法在CT图像重建中对于提高图像的重建质量和提高信噪比的可行性与有效性.另外还研究了字典学习中图像块大小和滑动距离对重建图像的影响     

限制步长法及其推广

本文给出一个新的限制步长算法并讨论了算法的收敛性质.考虑问题这里f(x)∈C~2,C是R~n中的闭凸集.对于给定集合K,实数h及点y,定义hK={hx|x∈K},y+K={y+x|x∈K},而(?)表示K之边界点集.1.限制步长算法算法Ⅰ任意取定     

Hilbert空间中求解分裂可行问题CQ算法的强收敛性

在Hilbert空间中,为了研究分裂可行问题迭代算法的强收敛性,提出了一种新的CQ算法.首先利用CQ算法构造了一个改进的Halpern迭代序列； 然后通过把分裂可行问题转化为算子不动点， 在较弱的条件下， 证明了该序列强收敛到分裂可行问题的一个解. 推广了Wang和Xu的有关结果.     

基于SCN函数共轭梯度方向的稀疏向量特征提取算法

稀疏向量特征提取是指在优化时利用各种范数对解进行约束,从而获得带有稀疏特征的最优解,其广泛应用于复杂系统中的机器学习、深度学习和大数据分析等领域的特征提取问题.大量的研究表明各种范数如L₀范数、L₁范数和L₂范数的方法都存在各自的缺点,主要表现在越容易求解的范数越不精准稀疏,越精准稀疏的范数越难求解.文章提出了一种基于SCN函数共轭梯度方向的稀疏向量特征发现算法(CGDL),稀疏向量特征发现可以用一个稀疏特征提取优化模型建立,其目标函数是一个SCN函数,对其中的L₀范数进行转换,形成一个具有特殊结构优化问题,这个问题等价于双层规划的凸-凹极小极大化问题,这类问题可以解决稀疏回归、图像特征和压缩感知等问题.文章给出了上述模型的稀疏特征提取算法的详细计算步骤和收敛性分析证明,并且对给定的实际数据集和高维模拟数据集对算法的有效性、复杂性和收敛速度进行了数值对比实验,表明了该算法在精准度和稀疏性上显著优于其他对比方法,并且具有较好的收敛速度.     

求解稀疏分裂可行问题的一种投影算法

本文研究了稀疏分裂可行问题.通过将分裂可行问题转化为一个目标函数为凸函数的稀疏约束优化问题,设计一种梯度投影算法来求解此问题,获得了算法产生的点列可以收敛到稀疏分裂可行问题的一个解.用数值例子说明了算法的有效性.     

多集分裂等式问题的逐次松弛投影算法

多集分裂等式问题是分裂可行性问题的拓展问题,在图像重建、语言处理、地震探测等实际问题中具有广泛的应用.为了解决这个问题,提出了逐次松弛投影算法,设计了变化的步长,使其充分利用当前迭代点的信息且不需要算子范数的计算,证明了算法的弱收敛性.数值算例验证了算法在迭代次数与运行时间等方面的优越性.     

A new self-adaptive CQ algorithm with an application to the LASSO problem

In this paper, we introduce a new self-adaptive CQ algorithm for solving split feasibility problems in real Hilbert spaces. The algorithm is designed, such that the stepsizes are directly computed at each iteration. We also consider the corresponding relaxed CQ algorithm for the proposed method. Under certain mild conditions, we establish weak convergence of the proposed algorithm as well as strong convergence of its hybrid-type variant. Finally, numerical examples illustrating the efficiency of our algorithm in solving the LASSO problem are presented.     

A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSFP) is to find a point belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. Many iterative methods can be employed to solve the MSFP. Jinling Zhao et al. proposed a modification for the CQ algorithm and a relaxation scheme for this modification to solve the MSFP. The strong convergence of these algorithms are guaranteed in finite-dimensional Hilbert spaces. Recently López et al. proposed a relaxed CQ algorithm for solving split feasibility problem, this algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, this algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed self-adaptive CQ algorithm for solving the MSFP where closed convex sets are level sets of some convex functions such that the strong convergence is guaranteed in the framework of infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results.     

Relaxed two points projection method for solving the multiple-sets split equality problem

The multiple-sets split equality problem, a generalization and extension of the split feasibility problem, has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing. It can be a model for many inverse problems where constraints are imposed on the solutions in the domains of two linear operators as well as in the operators’ ranges simultaneously. Although, for the split equality problem, there exist many algorithms, there are but few algorithms for the multiple-sets split equality problem. Hence, in this paper, we present a relaxed two points projection method to solve the problem; under some suitable conditions, we show the weak convergence and give a remark for the strong convergence method in the Hilbert space. The interest of our algorithm is that we transfer the problem to an optimization problem, then, based on the model, we present a modified gradient projection algorithm by selecting two different initial points in different sets for the problem (we call the algorithm as two points algorithm). During the process of iteration, we employ subgradient projections, not use the orthogonal projection, which makes the method implementable. Numerical experiments manifest the algorithm is efficient.     

Viscosity Approximation Methods for a Class of Generalized Split Feasibility Problems with Variational Inequalities in Hilbert Space

Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP’s with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space.     

Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications

In this paper, we present hybrid inertial proximal algorithms for the split variational inclusion problems in Hilbert spaces, and provide convergence theorems for the proposed algorithms. In fact, an inertial type algorithm was proposed as an acceleration process. As application, we study split minimization problem, split feasibility problem, relaxed split feasibility problem and linear inverse problem in real Hilbert spaces. Finally, numerical results are given for our main results.     

A new simultaneous subgradient projection algorithm for solving a multiple-sets split feasibility problem

In this paper, we present a simultaneous subgradient algorithm for solving the multiple-sets split feasibility problem. The algorithm employs two extrapolated factors in each iteration, which not only improves feasibility by eliminating the need to compute the Lipschitz constant, but also enhances flexibility due to applying variable step size. The convergence of the algorithm is proved under suitable conditions. Numerical results illustrate that the new algorithm has better convergence than the existing one.