The positive definite completion problem revisited

In [R. Grone, C.R. Johnson, E. Sa, H. Wolkowicz, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl. 58 (1984) 109-124] the positive definite (semi-) completion problem in which the underlying graph is chordal was solved. For the positive definite case, the process was constructive and the completion was obtained by completing the partial matrix an entry at a time. For the positive semidefinite case, they obtained completions of a particular sequence of partial positive definite matrices with the same underlying graph and noted that there is a convergent subsequence of these completions that converges to the desired completion. Here, in the chordal case, we provide a constructive solution, based entirely on matrix/graph theoretic methods, to the positive (semi-)definite completion problem. Our solution associates a specific tree (called the “clique tree” [C.R. Johnson, M. Lundquist, Matrices with chordal inverse zero-patterns, Linear and Multilinear Algebra 36 (1993) 1-17]) with the (chordal) graph of the given partial positive (semi-)definite matrix. This tree structure allows us to complete the matrix a “block at a time” as opposed to an “entry at a time” (as in Grone et al. (1984) for the positive definite case). In Grone et al. (1984), using complex analytic techniques, the completion for the positive definite case was shown to be the unique determinant maximizing completion and was shown to be the unique completion that has zeros in its inverse in the positions corresponding to the unspecified entries of the partial matrix. Here, we show the same using only matrix/graph theoretic tools.     

Minimizing the condition number of a positive definite matrix by completion

Summary. We consider the problem of minimizing the spectral condition number of a positive definite matrix by completion: \noindent where is an Hermitian positive definite matrix, a matrix and is a free Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solution of this problem we characterize the complete set of matrices that give the minimum condition number. Received October 15, 1993     

非负定阵元素的扰动和正定完备化

在保持非负定性不变的前提下,本文对矩阵每一元素容许多大的扰动作了进一步的研究, 将本文的结论和C．R．Johnson提出的部分正定阵的正定完备化进行比较,容易发现对已知的正定矩阵求扰动,本文的结论比用C．R.Johnson的正定完备化计算扰动形式上更简单,同时也给出了不同于C．R．Johnson的部分正定阵的正定完备化表示的另外一个公式,推出了这些正定完备化矩阵应具有的若干性质．     

A matrix-splitting method for symmetric affine second-order cone complementarity problems

The affine second-order cone complementarity problem (SOCCP) is a wide class of problems that contains the linear complementarity problem (LCP) as a special case. The purpose of this paper is to propose an iterative method for the symmetric affine SOCCP that is based on the idea of matrix splitting. Matrix-splitting methods have originally been developed for the solution of the system of linear equations and have subsequently been extended to the LCP and the affine variational inequality problem. In this paper, we first give conditions under which the matrix-splitting method converges to a solution of the affine SOCCP. We then present, as a particular realization of the matrix-splitting method, the block successive overrelaxation (SOR) method for the affine SOCCP involving a positive definite matrix, and propose an efficient method for solving subproblems. Finally, we report some numerical results with the proposed algorithm, where promising results are obtained especially for problems with sparse matrices.     

边界约束二次规划问题的分解方法

１．引言本文考虑如下边界约束的二次规划问题：其中ＱＥ＊"""是对称的,Ｃ,人。Ｅ＊"是给定的常数向量,且Ｚ＜。这类问题经常出现在偏微分方程,离散化的连续时间最优控制问题、线性约束的最小二乘问题、工程设计、或作为非线性规划方法中的序列子问题．因此具有特殊的重要性．本文提出求解问题（１．１）的分解方法．它类似求解线性代数方程组的选代法,它是对Ｑ进行正则分裂【对即把Ｑ分裂为两个矩阵之和,Ｑ＝Ｎ十片而这两个矩阵之差（Ｎ一则是对称正定的．在每次迭代中用一个易于求解的矩阵Ｎ替代Ｑ进行计算一新的二次规划问题．在适…     

An investigation of interior-point and block pivoting algorithms for large-scale symmetric monotone linear complementarity problems

In this paper we describe a computational study of block principal pivoting (BP) and interior-point predictor-corrector (PC) algorithms for the solution of large-scale linear complementarity problems (LCP) with symmetric positive definite matrices. This study shows that these algorithms are in general quite appropriate for this type of LCPs. The BP algorithm does not seem to be sensitive to bad scaling and degeneracy of the unique solution of the LCP, while these aspects have some effect on the performance of the PC algorithm. On the other hand, the BP method has not performed well in two LCPs with ill-conditioned matrices for which the PC algorithm has behaved quite well.A hybrid algorithm combining these two techniques is also introduced and seems to be the most robust procedure for the solution of large-scale LCPs with symmetric positive definite matrices.Support of this work has been provided by the Instituto de Telecomunicações.     

Efficient numerical methods in non-uniform sampling theory

Summary. We present a new ``second generation" reconstruction algorithm for irregular sampling, i\@.e\@. for the problem of recovering a band-limited function from its non-uniformly sampled values. The efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems. The choice of "adaptive weights" can be seen as a simple but very efficient method of preconditioning. Further substantial acceleration is achieved by utilizing the Toeplitz-type structure of the system matrix. This new algorithm can handle problems of much larger dimension and condition number than have been accessible so far. Furthermore, if some gaps between samples are large, then the algorithm can still be used as a very efficient extrapolation method across the gaps. Received December 22, 1993     

半监督度量学习内蕴最速下降算法的收敛性分析

主要研究对称正定矩阵群上的内蕴最速下降算法的收敛性问题.首先针对一个可转化为对称正定矩阵群上无约束优化问题的半监督度量学习模型,提出对称正定矩阵群上一种自适应变步长的内蕴最速下降算法.然后利用李群上的光滑函数在任意一点处带积分余项的泰勒展开式,证明所提算法在对称正定矩阵群上是线性收敛的.最后通过在分类问题中的数值实验说明算法的有效性.     

Local convergence of the (exact and inexact) iterative aggregation method for linear systems and Markov operators

Summary. The iterative aggregation method for the solution of linear systems is extended in several directions: to operators on Banach spaces; to the method with inexact correction, i.e., to methods where the (inner) linear system is in turn solved iteratively; and to the problem of finding stationary distributions of Markov operators. Local convergence is shown in all cases. Convergence results apply to the particular case of stochastic matrices. Moreover, an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled Markov chains, as well as for Markov chains with other zero-nonzero structures. Received May 25, 1991/Revised version received February 23, 1994     

Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations

Summary We discuss block matrices of the formA=[A _ij ], whereA _ij is ak×k symmetric matrix,A _ij is positive definite andA _ij is negative semidefinite. These matrices are natural block-generalizations of Z-matrices and M-matrices. Matrices of this type arise in the numerical solution of Euler equations in fluid flow computations. We discuss properties of these matrices, in particular we prove convergence of block iterative methods for linear systems with such system matrices.     

Application of a block modified Chebyshev algorithm to the iterative solution of symmetric linear systems with multiple right hand side vectors

Summary. An adaptive Richardson iteration method is described for the solution of large sparse symmetric positive definite linear systems of equations with multiple right-hand side vectors. This scheme ``learns' about the linear system to be solved by computing inner products of residual matrices during the iterations. These inner products are interpreted as block modified moments. A block version of the modified Chebyshev algorithm is presented which yields a block tridiagonal matrix from the block modified moments and the recursion coefficients of the residual polynomials. The eigenvalues of this block tridiagonal matrix define an interval, which determines the choice of relaxation parameters for Richardson iteration. Only minor modifications are necessary in order to obtain a scheme for the solution of symmetric indefinite linear systems with multiple right-hand side vectors. We outline the changes required. Received April 22, 1993     

Cuts, matrix completions and graph rigidity

This paper brings together several topics arising in distinct areas: polyhedral combinatorics, in particular, cut and metric polyhedra; matrix theory and semidefinite programming, in particular, completion problems for positive semidefinite matrices and Euclidean distance matrices; distance geometry and structural topology, in particular, graph realization and rigidity problems. Cuts and metrics provide the unifying theme. Indeed, cuts can be encoded as positive semidefinite matrices (this fact underlies the approximative algorithm for max-cut of Goemans and Williamson) and both positive semidefinite and Euclidean distance matrices yield points of the cut polytope or cone, after applying the functions 1/π arccos(.) or √. When fixing the dimension in the Euclidean distance matrix completion problem, we find the graph realization problem and the related question of unicity of realization, which leads to the question of graph rigidity. Our main objective here is to present in a unified setting a number of results and questions concerning matrix completion, graph realization and rigidity problems. These problems contain indeed very interesting questions relevant to mathematical programming and we believe that research in this area could yield to cross-fertilization between the various fields involved.     

A Riemannian submersion-based approach to the Wasserstein barycenter of positive definite matrices

In this paper, we introduce a novel geometrization on the space of positive definite matrices, derived from the Riemannian submersion from the general linear group to the space of positive definite matrices, resulting in easier computation of its geometric structure. The related metric is found to be the same as a particular Wasserstein metric. Based on this metric, the Wasserstein barycenter problem is studied. To solve this problem, some schemes of the numerical computation based on gradient descent algorithms are proposed and compared. As an application, we test the k-means clustering of positive definite matrices with different choices of matrix mean.     

A modified Quasi-Newton method for vector optimization problem

In this paper, existence of critical point and weak efficient point of vector optimization problem is studied. A sequence of points in n-dimension is generated using positive definite matrices like Quasi-Newton method. It is proved that accumulation points of this sequence are critical points or weak efficient points under different conditions. An algorithm is provided in this context. This method is free from any kind of priori chosen weighting factors or any other form of a priori ranking or ordering information for objective functions. Also, this method does not depend upon initial point. The algorithm is verified in numerical examples.     

On extremum properties of orthogonal quotients matrices

In this paper we explore the extremum properties of orthogonal quotients matrices. The orthogonal quotients equality that we prove expresses the Frobenius norm of a difference between two matrices as a difference between the norms of two matrices. This turns the Eckart-Young minimum norm problem into an equivalent maximum norm problem. The symmetric version of this equality involves traces of matrices, and adds new insight into Ky Fan’s extremum problems. A comparison of the two cases reveals a remarkable similarity between the Eckart-Young theorem and Ky Fan’s maximum principle. Returning to orthogonal quotients matrices we derive “rectangular” extensions of Ky Fan’s extremum principles, which consider maximizing (or minimizing) sums of powers of singular values.     

关于实方阵的几个问题

本文讨论如下内容：１．把有关对称正定（半正定）的一些性质推广到广义正定（半正定）。２．给定ｘ∈Ｒｍ×ｍ，∧为对角阵，求ＡＸ＝ｘ∧在对称半正定矩阵类中解存在的充要条件及一般形式，并讨论了对任意给定的对称正定（半正定）矩阵Ａ，在上述解的集合中求得Ａ，使得     

The geometric mean and B-loop quasigroups of the convex cone of positive definite matrices

In this paper we consider the convex cone of positive definite matrices as algebraic system equipped with geometric mean and B-loop from the standard matrix polar decomposition. Some algebraic structures of these quasigroups are investigated in the context of matrix theory. In particular, their autotopism groups are completely determined: they are isomorphic to the group of positive real numbers.Received: 28 April 2004     

Dykstra's Algorithm for a Constrained Least-squares Matrix Problem

We apply Dykstra's alternating projection algorithm to the constrained least-squares matrix problem that arises naturally in statistics and mathematical economics. In particular, we are concerned with the problem of finding the closest symmetric positive definite bounded and patterned matrix, in the Frobenius norm, to a given matrix. In this work, we state the problem as the minimization of a convex function over the intersection of a finite collection of closed and convex sets in the vector space of square matrices. We present iterative schemes that exploit the geometry of the problem, and for which we establish convergence to the unique solution. Finally, we present preliminary numberical results to illustrate the performance of the proposed iterative methods.     

Detecting hyperbolic and definite matrix polynomials

Hyperbolic or more generally definite matrix polynomials are important classes of Hermitian matrix polynomials. They allow for a definite linearization and can therefore be solved by a standard algorithm for Hermitian matrices. They have only real eigenvalues which can be characterized as minmax and maxmin values of Rayleigh functionals, but there is no easy way to test if a given polynomial is hyperbolic or definite or not. Taking advantage of the safeguarded iteration which converges globally and monotonically to extreme eigenvalues we obtain an efficient algorithm that identifies hyperbolic or definite polynomials and enables the transformation to an equivalent definite linear pencil. Numerical examples demonstrate the efficiency of the approach.     

A smoothing Newton's method for the construction of a damped vibrating system from noisy test eigendata

In this paper we consider an inverse problem for a damped vibration system from the noisy measured eigendata, where the mass, damping, and stiffness matrices are all symmetric positive‐definite matrices with the mass matrix being diagonal and the damping and stiffness matrices being tridiagonal. To take into consideration the noise in the data, the problem is formulated as a convex optimization problem involving quadratic constraints on the unknown mass, damping, and stiffness parameters. Then we propose a smoothing Newton‐type algorithm for the optimization problem, which improves a pre‐existing estimate of a solution to the inverse problem. We show that the proposed method converges both globally and quadratically. Numerical examples are also given to demonstrate the efficiency of our method. Copyright © 2008 John Wiley & Sons, Ltd.