凸可行问题的平行近似次梯度投影算法

对凸可行问题提出了包括上松弛的平行近似次梯度投影算法和加速平行近似次梯度投影算法．与序列近似次梯度投影算法相比, 平行近似次梯度投影算法(每次迭代同时运用多个凸集的近似次梯度超平面上的投影)能够保证迭代序列收敛到离各个凸集最近的点． 上松弛的迭代技术和含有外推因子的加速技术的应用, 减少了数据存储量, 提高了收 敛速度. 最后在较弱的条件下证明了算法的收敛性, 数值实验结果验证了算法的有效性和优越性.     

一般单调变分不等式的近似邻近外梯度算法

近似邻近点算法是求解单调变分不等式的一个有效方法,该算法通过解决一系列强单调子问题,产生近似邻近点序列来逼近变分不等式的解,而外梯度算法则通过每次迭代中增加一个投影来克服一般投影算法限制太强的缺点,但它们均未能改变迭代步骤中不规则闭凸区域上投影难计算的问题.于是,本文结合外梯度算法的迭代格式,构造包含原投影区域的半空间,将投影建立在半空间上,简化了投影的求解过程,并对新的邻近点序列作相应限制,使得改进的算法具有较好的收敛性.     

一类新的伪单调变分不等式的自适应次梯度外梯度投影算法

Gibali[J.Nonlinear Anal.Optim.,2015,6(1):41-51]提出了一种解伪单调非Lipschitz连续变分不等式的自适应次梯度外梯度投影算法.其下一迭代点是通过向一个特定的半空间投影来实施.本文通过构造新的下降方向得到了一类新的自适应次梯度外梯度投影算法,并借助于何炳生和廖立志[J.Optim.Theory Appl.,2002,112(1):111-128]中的技巧优化了这些算法的步长.证明了这些算法所生成序列的全局收敛性.数值实验结果表明这类次梯度外梯度投影算法比已有算法受初始点的选取、变分不等式的维数及停止标准的精度的影响更小.而且,从迭代次数及运算所花的时间来看,新的算法均优于Gibali提出的算法.     

广义非凸变分不等式解的存在性与投影算法

本文运用Banach压缩映象原理和投影技巧研究一类新的广义非凸变分不等式问题解的存在唯一性,并在非凸集上建立一个逼近广义非凸变分不等式解的三步投影算法,在一定条件下证明了该投影算法所产生的迭代序列的收敛性.     

一类新的广义非凸集值变分不等式组

本文的主要目的是引入一类广义非凸集值变分不等式.首先,我们把这类广义非凸集值变分不等式等价的转化为不动,点问题,通过构造一种新的扰动投影算法,在一定条件下,我们证明了所给迭代算法是收敛的.     

集值混合变分不等式的一类自适应惯性投影次梯度算法

本文在实Hilbert空间上引入了一类求解集值混合变分不等式新的自适应惯性投影次梯度算法.在集值映射T为f-强伪单调或单调的条件下,我们证明了由该自适应惯性投影次梯度算法所产生的序列强收敛于集值混合变分不等式问题的的唯一解.     

非线性规划的法向与梯度组合方向算法及其收敛性

求解上述非线性不等式约束的规划问题并使用梯度投影时,由于非线性约束的特性,目标函数的负梯度在迭代点所在的切平面的交上的投影方向不一定是可行方向.为了利用梯度投影求得一个可行的下降方向,并使算法具有收敛性质,往往需要不止一次的作投影计算,因而算法比较复杂.文献[1]一反以往需多次求投影来求得迭代方向的办法,首先采用斜投影以求迭代方向,使得计算减少到至多求两次投影并给出他的算法的收敛性     

闭凸集约束下线性矩阵方程求解的松弛交替投影算法

研究线性矩阵方程AXB=C在闭凸集合R约束下的数值迭代解法.所考虑的闭凸集合R为(1)有界矩阵集合,(2)Q-正定矩阵集合和(3)矩阵不等式解集合.构造松弛交替投影算法求解上述问题,并用算子理论证明了由该算法生成的序列具有弱收敛性.给出了矩阵方程AXB=C求对称非负解和对称半正定解的数值算例,大量数值实验验证了该算法的可行性和高效性,并说明该算法与交替投影算法和谱投影梯度算法比较在迭代效率上的明显优势.     

随机广义拟变分不等式的迭代解法及应用

为了获得Hilbert空间中一类随机广义拟变分不等式的迭代解法,证明了点到由具闭(凸)值的随机集值映射所刻画的变约束集上的投影算子的可测性.利用该可测性结果和可测选择定理,构造了求解随机广义拟变分不等式的随机迭代算法.在单调性及Lipschitz连续性条件下,获得了由算法生成的随机序列的收敛性.作为应用,给出了随机广义Nash博弈和随机Walrasian均衡问题的一些刻画性结果.     

一种新的求解变分不等式的惯性双次梯度外梯度算法（英文）

当可行集为一光滑凸函数的下水平集时,文献[Optimization,2020,69(6):1237-1253]提出了一种惯性双次梯度外梯度算法来求解Hilbert空间中的单调且Lipschitz连续的变分不等式问题.该算法在每次迭代中仅需向一个半空间计算两次投影,并得到了算法的弱收敛结果.本文通过使用黏性方法以及在惯性步采用新的步长来修正该算法.在适当的假设条件下证明了新算法所生成的序列能强收敛到变分不等式的一个解.此外,新算法在每次迭代中也仅需向半空间计算两次投影.     

Extrapolation algorithm for affine-convex feasibility problems

The convex feasibility problem under consideration is to find a common point of a countable family of closed affine subspaces and convex sets in a Hilbert space. To solve such problems, we propose a general parallel block-iterative algorithmic framework in which the affine subspaces are exploited to introduce extrapolated over-relaxations. This framework encompasses a wide range of projection, subgradient projection, proximal, and fixed point methods encountered in various branches of applied mathematics. The asymptotic behavior of the method is investigated and numerical experiments are provided to illustrate the benefits of the extrapolations.     

Projected subgradient methods with non-Euclidean distances for non-differentiable convex minimization and variational inequalities

We study subgradient projection type methods for solving non-differentiable convex minimization problems and monotone variational inequalities. The methods can be viewed as a natural extension of subgradient projection type algorithms, and are based on using non-Euclidean projection-like maps, which generate interior trajectories. The resulting algorithms are easy to implement and rely on a single projection per iteration. We prove several convergence results and establish rate of convergence estimates under various and mild assumptions on the problem’s data and the corresponding step-sizes. We dedicate this paper to Boris Polyak on the occasion of his 70th birthday.     

Subgradient Projection Algorithms and Approximate Solutions of Convex Feasibility Problems

In the present paper, we use subgradient projection algorithms for solving convex feasibility problems. We show that almost all iterates, generated by a subgradient projection algorithm in a Hilbert space, are approximate solutions. Moreover, we obtain an estimate of the number of iterates which are not approximate solutions. In a finite-dimensional case, we study the behavior of the subgradient projection algorithm in the presence of computational errors. Provided computational errors are bounded, we prove that our subgradient projection algorithm generates a good approximate solution after a certain number of iterates.     

An approximate subgradient algorithm for unconstrained nonsmooth,nonconvex optimization

In this paper a new algorithm for minimizing locally Lipschitz functions is developed. Descent directions in this algorithm are computed by solving a system of linear inequalities. The convergence of the algorithm is proved for quasidifferentiable semismooth functions. We present the results of numerical experiments with both regular and nonregular objective functions. We also compare the proposed algorithm with two different versions of the subgradient method using the results of numerical experiments. These results demonstrate the superiority of the proposed algorithm over the subgradient method.      

解凸可行问题的一个带中心技术的外推平行次梯度投影算法（英文）

In this paper,we present an extrapolated parallel subgradient projection method with the centering technique for the convex feasibility problem,the algorithm improves the convergence by reason of using centering techniques which reduce the oscillation of the corresponding sequence.To prove the convergence in a simply way,we transmit the parallel algorithm in the original space to a sequential one in a newly constructed product space.Thus,the convergence of the parallel algorithm is derived with the help of the sequential one under some suitable conditions.Numerical results show that the new algorithm has better convergence than the existing algorithms.     

Adaptive Projected Subgradient Method for Asymptotic Minimization of Sequence of Nonnegative Convex Functions

Abstract

This paper presents an algorithm, named adaptive projected subgradient method that can minimize asymptotically a certain sequence of nonnegative convex functions over a closed convex set in a real Hilbert space. The proposed algorithm is a natural extension of the Polyak's subgradient algorithm, for nonsmooth convex optimization problem with a fixed target value, to the case where the convex objective itself keeps changing in the whole process. The main theorem, showing the strong convergence of the algorithm as well as the asymptotic optimality of the sequence generated by the algorithm, can serve as a unified guiding principle of a wide range of set theoretic adaptive filtering schemes for nonstationary random processes. These include not only the existing adaptive filtering techniques; e.g., NLMS, Projected NLMS, Constrained NLMS, APA, and Adaptive parallel outer projection algorithm etc., but also new techniques; e.g., Adaptive parallel min-max projection algorithm, and their embedded constraint versions. Numerical examples show that the proposed techniques are well-suited for robust adaptive signal processing problems.     

A projection subgradient method for solving optimization with variational inequality constraints

In this paper, we propose a projection subgradient method for solving some classical variational inequality problem over the set of solutions of mixed variational inequalities. Under the conditions that $T$ is a $\Theta $ -pseudomonotone mapping and $A$ is a $\rho $ -strongly pseudomonotone mapping, we prove the convergence of the algorithm constructed by projection subgradient method. Our algorithm can be applied for instance to some mathematical programs with complementarity constraints.     

Bi-extrapolated subgradient projection algorithm for solving multiple-sets split feasibility problem

This paper deals with a bi-extrapolated subgradient projection algorithm by intro- ducing two extrapolated factors in the iterative step to solve the multiple-sets split feasibility problem. The strategy is intend to improve the convergence. And its convergence is proved un- der some suitable conditions. Numerical results illustrate that the bi-extrapolated subgradient projection algorithm converges more quickly than the existing algorithms.     

A general iterative scheme with applications to convex optimization and related fields

A general iterative scheme including relaxation and a corresponding problem class are presented. Some global convergence results are given. The acceleration of convergence is discussed, The scheme comprises a lot of known iterative methods such as subgradient methods and methods of successive orthogonal projections with relaxation. Applications to convex optimization, convex feasibility problems, systems of convex inequalities, variational inequalities, operator equations and systems of linear equations are given.