Numerical methods for solving some matrix feasibility problems

In this paper, we design two numerical methods for solving some matrix feasibility problems, which arise in the quantum information science. By making use of the structured properties of linear constraints and the minimization theorem of symmetric matrix on manifold, the projection formulas of a matrix onto the feasible sets are given, and then the relaxed alternating projection algorithm and alternating projection algorithm on manifolds are designed to solve these problems. Numerical examples show that the new methods are feasible and effective.     

Matrix pencils and existence conditions for quadratic programming with a sign-indefinite quadratic equality constraint

We consider minimization of a quadratic objective function subject to a sign-indefinite quadratic equality constraint. We derive necessary and sufficient conditions for the existence of solutions to the constrained minimization problem. These conditions involve a generalized eigenvalue of the matrix pencil consisting of a symmetric positive-semidefinite matrix and a symmetric indefinite matrix. A complete characterization of the solution set to the constrained minimization problem in terms of the eigenspace of the matrix pencil is provided.     

Symmetric minimum-norm updates for use in gibbs free energy calculations

This paper examines a type of symmetric quasi-Newton update for use in nonlinear optimization algorithms. The updates presented here impose additional properties on the Hessian approximations that do not result if the usual quasi-Newton updating schemes are applied to certain Gibbs free energy minimization problems. The updates derived in this paper are symmetric matrices that satisfy a given matrix equation and are least squares solutions to the secant equation. A general representation for this class of updates is given. The update in this class that has the minimum weighted Frobenius norm is also presented. This work was done at Sandia National Laboratories and supported by the US Dept. of Energy under contract no. DE-AC04-76DP00789.     

Subgradient methods for huge-scale optimization problems

We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piece-wise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, which total cost depends logarithmically in the dimension. This technique is based on a recursive update of the results of matrix/vector products and the values of symmetric functions. It works well, for example, for matrices with few nonzero diagonals and for max-type functions. We show that the updating technique can be efficiently coupled with the simplest subgradient methods, the unconstrained minimization method by B.Polyak, and the constrained minimization scheme by N.Shor. Similar results can be obtained for a new nonsmooth random variant of a coordinate descent scheme. We present also the promising results of preliminary computational experiments.     

Level‐based heuristics and hill climbing for the antibandwidth maximization problem

The antibandwidth maximization problem aims to maximize the minimum distance of entries of a sparse symmetric matrix from the diagonal and as such may be regarded as the dual of the well‐known bandwidth minimization problem. In this paper, we consider the feasibility of adapting heuristic algorithms for the bandwidth minimization problem to the antibandwidth maximization problem. In particular, using an inexpensive level‐based heuristic, we obtain an initial ordering that we refine using a hill‐climbing algorithm. This approach performs well on matrices coming from a range of practical problems with an underlying mesh. Comparisons with existing approaches show that, on this class of problems, our algorithm can be competitive with recently reported results in terms of quality while being significantly faster and applicable to much larger problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Structure methods for solving the nearest correlation matrix problem

The nearest correlation matrix problem is to find a positive semidefinite matrix with unit diagonal, that is, nearest in the Frobenius norm to a given symmetric matrix A. This problem arises in the finance industry, where the correlations are between stocks. In this paper, we formulate this problem as a smooth unconstrained minimization problem, for which rapid convergence can be obtained. Other methods are also studied. Comparative numerical results are reported.     

Partial symmetry of least energy nodal solutions to some variational problems

We investigate the symmetry properties of several radially symmetric minimization problems. The minimizers which we obtain are nodal solutions of superlinear elliptic problems, or eigenfunctions of weighted asymmetric eigenvalue problems, or they lie on the first curve in the Fucik spectrum. In all instances, we prove that the minimizers are foliated Schwarz symmetric. We give examples showing that the minimizers are in general not radially symmetric. The basic tool which we use is polarization, a concept going back to Ahlfors. We develop this method of symmetrization for sign changing functions. Supported by NATO grant PST.CLG.978736. Supported by DFG grant WE 2821/2-1. Supported by NATO grant SPT.CLG.978736.     

Low Tucker rank tensor completion using a symmetric block coordinate descent method

Low Tucker rank tensor completion has wide applications in science and engineering. Many existing approaches dealt with the Tucker rank by unfolding matrix rank. However, unfolding a tensor to a matrix would destroy the data's original multi-way structure, resulting in vital information loss and degraded performance. In this article, we establish a relationship between the Tucker ranks and the ranks of the factor matrices in Tucker decomposition. Then, we reformulate the low Tucker rank tensor completion problem as a multilinear low rank matrix completion problem. For the reformulated problem, a symmetric block coordinate descent method is customized. For each matrix rank minimization subproblem, the classical truncated nuclear norm minimization is adopted. Furthermore, temporal characteristics in image and video data are introduced to such a model, which benefits the performance of the method. Numerical simulations illustrate the efficiency of our proposed models and methods.     

Concave gauge functions and applications

Many problems of optimization involve the minimization of an objective function on a convex cone. In this respect we define a concave gauge function which will be used in interior point methods.Application are given in particular on the space of real symmetric matrices.     

AN ITERATED-SUBSPACE MINIMIZATION METHODS WITH SYMMETRIC RANK-ONE UPDATING

We consider an Iterated-Subspace Minimization(ISM) method for solving large-scale unconstrained minimization problems.At each major iteration of the method, a two-dimensional manifold, the iterated subspace, is constructed and an approximate minimizer of the objective function in this manifold then determined, and a symmetric rank-one updating is used to solve the inner minimization problem.     

An incomplete Hessian Newton minimization method and its application in a chemical database problem

To efficiently solve a large scale unconstrained minimization problem with a dense Hessian matrix, this paper proposes to use an incomplete Hessian matrix to define a new modified Newton method, called the incomplete Hessian Newton method (IHN). A theoretical analysis shows that IHN is convergent globally, and has a linear rate of convergence with a properly selected symmetric, positive definite incomplete Hessian matrix. It also shows that the Wolfe conditions hold in IHN with a line search step length of one. As an important application, an effective IHN and a modified IHN, called the truncated-IHN method (T-IHN), are constructed for solving a large scale chemical database optimal projection mapping problem. T-IHN is shown to work well even with indefinite incomplete Hessian matrices. Numerical results confirm the theoretical results of IHN, and demonstrate the promising potential of T-IHN as an efficient minimization algorithm.     

Convergence of Fixed-Point Continuation Algorithms for Matrix Rank Minimization

The matrix rank minimization problem has applications in many fields, such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixed-point continuation algorithm for solving the nuclear norm minimization problem (Math. Program., doi:, 2009). By incorporating an approximate singular value decomposition technique in this algorithm, the solution to the matrix rank minimization problem is usually obtained. In this paper, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving affinely constrained matrix rank minimization problems are reported.     

实对称带状矩阵逆特征值问题

研究了一类实对称带状矩阵逆特征值问题：给定三个互异实数λ,μ和v及三个非零实向量x,y和z,分别构造实对称五对角矩阵T和实对称九对角矩阵A,使其都具有特征对(λ,x),(μ,y)和(v,z)．给出了此类问题的两种提法,研究了问题的可解性以及存在惟一解的充分必要条件,最后给出了数值算法和数值例子．     

Multilevel approaches for FSAI preconditioning

Factorized sparse approximate inverse (FSAI) preconditioners are robust algorithms for symmetric positive matrices, which are particularly attractive in a parallel computational environment because of their inherent and almost perfect scalability. Their parallel degree is even redundant with respect to the actual capabilities of the current computational architectures. In this work, we present two new approaches for FSAI preconditioners with the aim of improving the algorithm effectiveness by adding some sequentiality to the native formulation. The first one, denoted as block tridiagonal FSAI, is based on a block tridiagonal factorization strategy, whereas the second one, domain decomposition FSAI, is built by reordering the matrix graph according to a multilevel k‐way partitioning method followed by a bandwidth minimization algorithm. We test these preconditioners by solving a set of symmetric positive definite problems arising from different engineering applications. The results are evaluated in terms of performance, scalability, and robustness, showing that both strategies lead to faster convergent schemes regarding the number of iterations and total computational time in comparison with the native FSAI with no significant loss in the algorithmic parallel degree.     

Vector and matrix apportionment problems and separable convex integer optimization

The problems of (bi-)proportional rounding of a nonnegative vector or matrix, resp., are written as particular separable convex integer minimization problems. Allowing any convex (separable) objective function we use the notions of vector and matrix apportionment problems. As a broader class of problems we consider separable convex integer minimization under linear equality restrictions Ax = b with any totally unimodular coefficient matrix A. By the total unimodularity Fenchel duality applies, despite the integer restrictions of the variables. The biproportional algorithm of Balinski and Demange (Math Program 45:193–210, 1989) is generalized and derives from the dual optimization problem. Also, a primal augmentation algorithm is stated. Finally, for the smaller class of matrix apportionment problems we discuss the alternating scaling algorithm, which is a discrete variant of the well-known Iterative Proportional Fitting procedure.     

On vector and matrix median computation

The aim of this paper is to gain more insight into vector and matrix medians and to investigate algorithms to compute them. We prove relations between vector and matrix means and medians, particularly regarding the classical structure tensor. Moreover, we examine matrix medians corresponding to different unitarily invariant matrix norms for the case of symmetric 2×2 matrices, which frequently arise in image processing. Our findings are explained and illustrated by numerical examples. To solve the corresponding minimization problems, we propose several algorithms. Existing approaches include Weiszfeld’s algorithm for the computation of ?₂ vector medians and semi-definite programming, in particular, second order cone programming, which has been used for matrix median computation. In this paper, we adapt Weiszfeld’s algorithm for our setting and show that also two splitting methods, namely the alternating direction method of multipliers and the parallel proximal algorithm, can be applied for generalized vector and matrix median computations. Besides, we compare the performance of these algorithms numerically and apply them within local median filters.     

An implementable proximal point algorithmic framework for nuclear norm minimization

The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primal-dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner sub-problems are approximately solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state-of-the-art algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature.     

A penalty method for rank minimization problems in symmetric matrices

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal solution is a KKT point. We develop a penalty formulation of the problem. We present calmness results for locally optimal solutions to the penalty formulation. We also develop a proximal alternating linearized minimization (PALM) scheme for the penalty formulation, and investigate the incorporation of a momentum term into the algorithm. Computational results are presented.     

Analysis of Nonlinear Quadratic Control in Infinite Time

This study refers to minimization of quadratic functionals in infinite time. The coefficients of the quadratic form are quadratic matrices, function of the state variable. Dynamic constraints are represented by a bilinear differential systems of the form. The necessary extremum conditions determine the adjoint variables λ and the control variables u as functions of state variable, respectively the adjoint system corresponding to those functions. Thus it will be obtained a matrix differential equation where the solution representing the positive defined symmetric matrix P ( x ), verifies the Riccati algebraic equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity

In this paper, we consider the nonlinear fractional Schrödinger equations with Hartree type nonlinearity. We obtain the existence of standing waves by studying the related constrained minimization problems via applying the concentration-compactness principle. By symmetric decreasing rearrangements, we also show that the standing waves, up to translations and phases, are positive symmetric nonincreasing functions. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near it for all time.