一类带凸约束的单调非线性方程组的无导数投影法收敛率分析

在适当的条件下,本文进一步研究带凸约束的单调非线性方程组的超记忆梯度型方法,并建立了它的次线性收敛率.同时,本文还给出求解带凸约束的单调非线性方程组的无导数投影法的更一般的算法框架,并分析了其收敛性及其收敛率.两个说明性的数值计算实例表明了该算法框架的可行性.     

解凸约束非线性单调方程组的无导数谱PRP投影算法

本文在著名PRP共轭梯度算法的基础上研究了一种无导数谱PRP投影算法,并证明了算法在求解带有凸约束条件的非线性单调方程组问题的全局收敛性.由于无导数和储存量小的特性,它更适应于求解大规模非光滑的非线性单调方程组问题.数值试验表明,新算法对给定的测试问题是有效的和稳定的.     

谱HS投影算法求解非线性单调方程组

借助谱梯度法和HS共轭梯度法的结构, 建立一种求解非线性单调方程组问题的谱HS投影算法. 该算法继承了谱梯度法和共轭梯度法储存量小和计算简单的特征, 且不需要任何导数信息, 因此它适应于求解大规模非光滑的非线性单调方程组问题. 在适当的条件下, 证明了该算法的收敛性, 并通过数值实验表明了该算法的有效性.     

求解非线性单调方程组的一种无导数投影算法

推广了一种修正的CG_DESCENT共轭梯度方法,并建立了一种有效求解非线性单调方程组问题的无导数投影算法.在适当的线搜索条件下,证明了算法的全局收敛性.由于新算法不需要借助任何导数信息,故它适应于求解大规模非光滑的非线性单调方程组问题.大量的数值试验表明,新算法对给定的测试问题是有效的.     

非线性单调方程组的三项无导数投影算法

本文讨论了一种求解非线性单调方程组问题的三项无导数投影算法,并在适当的条件下证明了算法的全局收敛性和R-线性收敛速度.由于无需利用任何导数信息,该算法适合求解大规模的非线性单调方程组问题.数值比较表明该算法是有效的.     

解凸约束非线性单调方程组的无导数低存储Broyden族投影法

拟牛顿法是求解非线性方程组的一类有效方法.相较于经典的牛顿法,拟牛顿法不需要计算Jacobian矩阵且仍具有超线性收敛性.本文基于BFGS和DFP的迭代公式,构造了新的充分下降方向.将该搜索方向和投影技术相结合,本文提出了无导数低存储的投影算法求解带凸约束的非线性单调方程组并证明了该算法是全局且R-线性收敛的.最后,将该算法用于求解压缩感知问题.实验结果表明,本文所提出的算法具有良好的计算效率和稳定性.     

凸约束非光滑方程组一个新的谱梯度投影算法

基于寻找分离超平面的三种经典线搜索技术,本文提出了一种自适应线搜索技术.结合谱梯度投影法,提出了凸约束非光滑单调方程组的一个谱梯度投影算法.该算法不需要计算和存储任何矩阵,因而适合求解大规模非光滑的非线性单调方程组.在较弱的条件下,证明了方法的全局收敛性,并分析了算法的收敛率.数值试验结果表明算法是有效的和鲁棒的.     

无约束最优化的一个修正的类BFGS算法

利用前一步得到的曲率信息代替xk到xk+1段二次模型的曲率给出一个具有和BFGS类似的收敛性质的类BFGS算法,并揭示新算法与自调比拟牛顿法的关系.从试验函数库CUTE中选择标准试验函数,对比标准BFGS算法及其它改进BFGS算法进行数值试验.试验结果表明这个新算法的表现有点象自调比拟牛顿算法.     

强单调对称非线性方程组的BFGS算法

提出一种求解强单调非线性方程组的BFGS算法,该算法的一个明显优点是Bκ的条件数比Li-Fukushima^[3]提出的GNBFGS中Bκ的条件数小得多。且该算法是一种无需计算导数的下降算法。在一定的条件下,证明了算法的全局收敛性和超线性收敛性。最后进行数值试验,结果表明,本文算法具有较好的数值结果。而且验证了本文所提出的算法中Bκ的条件数要比GNBFGS算法的条件数小得多。     

非线性伪单调方程组的谱LS型投影算法

基于谱梯度法和著名LS共轭梯度法的结构,该文建立了求解凸约束非线性伪单调方程组问题的谱LS型无导数投影算法.通过构建适当的谱参数,该算法在每一次迭代中都能保证搜索方向的充分下降性,并且独立于线搜索条件.在适当的假设条件和经典无导数线搜索条件下,算法具有全局收敛性.通过数值实验发现,该算法继承了LS共轭梯度法优秀的计算性能,并提高了稳定性.     

Symmetric low‐rank corrections to quadratic models

In this paper, we study the quadratic model updating problems by using symmetric low‐rank correcting, which incorporates the measured model data into the analytical quadratic model to produce an adjusted model that matches the experimental model data, and minimizes the distance between the analytical and updated models. We give a necessary and sufficient condition on the existence of solutions to the symmetric low‐rank correcting problems under some mild conditions, and propose two algorithms for finding approximate solutions to the corresponding optimization problems. The good performance of the two algorithms is illustrated by numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.     

Solving inverse eigenvalue problems via Householder and rank-one matrices

A brief and practical algorithm is introduced to solve symmetric inverse eigenvalue problems, which we call HROU algorithm. The algorithm is based on Householder transformations and rank one updating. We give some basic properties and the computational amount and develop sensitivity analysis of HROU algorithm. Furthermore, we develop HROU algorithm into a multi-level and adaptive one, named MLAHROU, to solve symmetric nonnegative inverse eigenvalue problems. New sufficient conditions to ensure symmetric nonnegative matrices and symmetric M-matrices are given. Many numerical examples are given to verify our theory, compare with existing results and show the efficiency of our algorithms.     

Ranking fuzzy quantities based on the angle of the reference functions

Ordering fuzzy quantities and their comparison play a key tool in many applied models in the world and in particular decision-making procedures. However a huge number of researches is attracted to this filed but until now there is any unique accepted method to rank the fuzzy quantities. In fact, each proposed method may has some shortcoming. So we are going to present a novel method based on the angle of the reference functions to cover a wide range of fuzzy quantities by over coming the draw backs of some existing methods. In the mentioned firstly, the angle between the left and right membership functions (the reference functions) of every fuzzy set is called Angle of Fuzzy Set (AFS), and then in order to extend ranking of two fuzzy sets the angle of fuzzy sets and α-cuts is used. The method is illustrated by some numerical examples and in particular the results of ranking by the proposed method and some common and existing methods for ranking fuzzy sets is compared to verify the advantage of the new approach. In particular, based on the results of comparison of our method with well known methods which are exist in the literature, we will see that against of most existing ranking approaches, our proposed approach can rank fuzzy numbers that have the same mode and symmetric spreads. In fact, the proposed method in this paper can effectively rank symmetric fuzzy numbers as well as the effective methods which are appeared in the literature. Moreover, unlike of most existing ranking approaches, our proposed approach can rank non-normal fuzzy sets. Finally, we emphasize that the concept of fuzzy ordering is one of key role in establishing the numerical algorithms in operations research such as fuzzy primal simplex algorithms, fuzzy dual simplex algorithms and as well as discussed in the works of Ebrahimnejad and Nasseri and coworkers , , , , ,  and .     

Symmetric and Non-Symmetric ABS Methods for Solving Diophantine Systems of Equations

In the present paper we describe a new class of algorithms for solving Diophantine systems of equations in integer arithmetic. This algorithm, designated as the integer ABS (iABS) algorithm, is based on the ABS methods in the real space, with extensive modifications to ensure that all calculations remain in the integer space. Importantly, the iABS solves Diophantine systems of equations without determining the Hermite normal form. The algorithm is suitable for solving determined, over- or underdetermined, full rank or rank deficient linear integer equations. We also present a scaled integer ABS system and two special cases for solving general Diophantine systems of equations. In the scaled symmetric iABS (ssiABS), the Abaffian matrix H _i is symmetric, allowing that only half of its elements need to be calculated and stored. The scaled non-symmetric iABS system (snsiABS) provides more freedom in selecting the arbitrary parameters and thus the maximal values of H _i can be maintained at a certainly lower level. In addition to the above theoretical results, we also provide numerical experiments to test the performance of the ssiABS and the snsiABS algorithms. These experiments have confirmed the suitability of the iABS system for practical applications.     

RankRev: aMatlab package for computing the numerical rank and updating/downdating

The numerical rank determination frequently occurs in matrix computation when the conventional exact rank of a hidden matrix is desired to be recovered. This paper presents a Matlab package RankRev that implements two efficient algorithms for computing the numerical rank and numerical subspaces of a matrix along with updating/downdating capabilities for making adjustment to the results when a row or column is inserted/deleted. The package and the underlying algorithms are accurate, reliable, and much more efficient than the singular value decomposition when the matrix is of low rank or low nullity.     

Rank and null space calculations using matrix decomposition without column interchanges

The most widely used stable methods for numerical determination of the rank of a matrix A are the singular value decomposition and the QR algorithm with column interchanges. Here two algorithms are presented which determine rank and nullity in a numerically stable manner without using column interchanges. One algorithm makes use of the condition estimator of Cline, Moler, Stewart, and Wilkinson and relative to alternative stable algorithms is particularly efficient for sparse matrices. The second algorithm is important in the case that one wishes to test for rank and nullity while sequentially adding columns to a matrix.     

Marked length rigidity of rank one symmetric spaces and their product

In this paper we show that if two Zariski dense representations, from a group G into Iso(X) where X is rank one symmetric space, have the proportional marked length spectrum, then they are conjugate. As a generalization we show that a Zariski dense representation into the isometry group of the product of rank one symmetric spaces is determined by the marked cross ratio.     

Characterisation of multiple conducting permeable objects in metal detection by polarizability tensors

Realistic applications in metal detection involve multiple inhomogeneous‐conducting permeable objects, and the aim of this paper is to characterise such objects by polarizability tensors. We show that, for the eddy current model, the leading order terms for the perturbation in the magnetic field, due to the presence of N small conducting permeable homogeneous inclusions, comprises of a sum of N terms with each containing a complex symmetric rank 2 polarizability tensor. Each tensor contains information about the shape and material properties of one of the objects and is independent of its position. The asymptotic expansion we obtain extends a previously known result for a single isolated object and applies in situations where the object sizes are small and the objects are sufficiently well separated. We also obtain a second expansion that describes the perturbed magnetic field for inhomogeneous and closely spaced objects, which again characterises the objects by a complex symmetric rank 2 tensor. The tensor's coefficients can be computed by solving a vector valued transmission problem, and we include numerical examples to illustrate the agreement between the asymptotic formula describing the perturbed fields and the numerical prediction. We also include algorithms for the localisation and identification of multiple inhomogeneous objects.     

Fast algorithms for hierarchically semiseparable matrices

Semiseparable matrices and many other rank‐structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We represent HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS forms. Fast HSS algorithms including new HSS structure generation and HSS form Cholesky factorization are developed. Moreover, we provide a new linear complexity explicit ULV factorization algorithm for symmetric positive definite HSS matrices with a low‐rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the algorithms. All these algorithms have nice data locality. They are useful in developing fast‐structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc. Some applications are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

Updating the Self-Scaling Symmetric Rank One Algorithm with Limited Memory for Large-Scale Unconstrained Optimization

A new limited memory quasi-Newton algorithm is developed, in which the self-scaling symmetric rank one update with Davidon's optimal condition is applied. Preliminary numerical tests show that the new algorithm is very efficient for large-scale problems as well as general nonlinear optimization.