scholarly journals The maximal angle between 5×5 positive semidefinite and 5×5 nonnegative matrices

2021 ◽  
Vol 37 ◽  
pp. 698-708
Author(s):  
Qinghong Zhang
Keyword(s):  
Optimization Problem ◽  
Nonnegative Matrix ◽  
Positive Semidefinite ◽  
Problem Solver ◽  
Nonnegative Matrices ◽  
Geometric Program ◽  
Maximal Angle ◽  
Method Of Lagrange Multipliers ◽  
Semidefinite Matrix ◽  
Signomial Geometric Programming

The paper is devoted to the study of the maximal angle between the $5\times 5$ semidefinite matrix cone and $5\times 5$ nonnegative matrix cone. A signomial geometric programming problem is formulated in the process to find the maximal angle. Instead of using an optimization problem solver to solve the problem numerically, the method of Lagrange Multipliers is used to solve the signomial geometric program, and therefore, to find the maximal angle between these two cones.

scholarly journals On the maximal angle between copositive matrices

2014 ◽  
Vol 27 ◽  
Author(s):  
Felix Goldberg ◽  
Naomi Shaked-Monderer
Keyword(s):  
Graph Theory ◽  
Variational Approach ◽  
Related Problem ◽  
Nonnegative Matrix ◽  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Algebraic Graph Theory ◽  
Copositive Matrices ◽  
Maximal Angle ◽  
Semidefinite Matrix

Hiriart-Urruty and Seeger have posed the problem of finding the maximal possible angle θ_{max}(C_n) between two copositive matrices of order n [J.-B. Hiriart-Urruty and A. Seeger. A variational approach to copositive matrices. SIAM Rev., 52:593–629, 2010.]. They have proved that θ_{max}(C_2) = (3/4)pi and conjectured that θ_{max}(C_n) is equal to (3/4)pi for all n ≥ 2. In this note, their conjecture is disproven by showing that lim_{n→∞} θ_{max}(C_n) = pi. The proof uses a construction from algebraic graph theory. The related problem of finding the maximal angle between a nonnegative matrix and a positive semidefinite matrix of the same order is considered in this paper.

scholarly journals The triangle graph $T_6$ is not SPN

2020 ◽  
Vol 36 (36) ◽  
pp. 90-93
Author(s):  
Stephen Drury
Keyword(s):  
Symmetric Matrix ◽  
Nonnegative Matrix ◽  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Real Symmetric Matrix ◽  
Triangle Graph ◽  
Nonnegative Vector ◽  
Copositive Matrix ◽  
Semidefinite Matrix

A real symmetric matrix $A$ is copositive if $x'Ax \geq 0$ for every nonnegative vector $x$. A matrix is SPN if it is a sum of a real positive semidefinite matrix and a nonnegative matrix. Every SPN matrix is copositive, but the converse does not hold for matrices of order greater than $4$. A graph $G$ is an SPN graph if every copositive matrix whose graph is $G$ is SPN. We show that the triangle graph $T_6$ is not SPN.

scholarly journals A trace bound for integer-diagonal positive semidefinite matrices

2020 ◽  
Vol 8 (1) ◽  
pp. 14-16
Author(s):  
Lon Mitchell
Keyword(s):  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Positive Semidefinite Matrices ◽  
Semidefinite Matrix

AbstractWe prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.

Parallel Nonnegative Matrix Factorization via Newton Iteration

2016 ◽  
Vol 26 (03) ◽  
pp. 1650014 ◽  
Author(s):  
Markus Flatz ◽  
Marián Vajteršic
Keyword(s):  
Shared Memory ◽  
Matrix Factorization ◽  
Message Passing ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Newton Iteration ◽  
Parallel Execution ◽  
Kkt Conditions ◽  
Nonnegative Matrices ◽  
First Order

The goal of Nonnegative Matrix Factorization (NMF) is to represent a large nonnegative matrix in an approximate way as a product of two significantly smaller nonnegative matrices. This paper shows in detail how an NMF algorithm based on Newton iteration can be derived using the general Karush-Kuhn-Tucker (KKT) conditions for first-order optimality. This algorithm is suited for parallel execution on systems with shared memory and also with message passing. Both versions were implemented and tested, delivering satisfactory speedup results.

Stability indicatrices of nonnegative matrices and some of their applications in problems of biology and epidemiology

2019 ◽  
Vol 34 (5) ◽  
pp. 269-275
Author(s):  
Valery N. Razzhevaikin
Keyword(s):  
Nonnegative Matrix ◽  
Host Population ◽  
Nonnegative Matrices ◽  
Biological Communities ◽  
Evolutionary Selection ◽  
Evolutionary Optimality ◽  
Selection For ◽  
The Stability ◽  
Series Of Functions

Abstract The method of constructing a stability indicatrix of a nonnegative matrix having the form of a polynomial of its coefficients is presented. The algorithm of construction and conditions of its applicability are specified. The applicability of the algorithm is illustrated on examples of constructing the stability indicatrix for a series of functions widely used in simulation of the dynamics of discrete biological communities, for solving evolutionary optimality problems arising in biological problems of evolutionary selection, for identification of the conditions of the pandemic in a distributed host population.

scholarly journals Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Li Wang
Keyword(s):  
Riccati Equation ◽  
Positive Semidefinite ◽  
Algebraic Riccati Equation ◽  
Positive Semidefinite Matrix ◽  
Matrix Inequalities ◽  
Numerical Examples ◽  
Upper Solution ◽  
Solution Bound ◽  
Semidefinite Matrix ◽  
Two Parameter

The continuous coupled algebraic Riccati equation (CCARE) has wide applications in control theory and linear systems. In this paper, by a constructed positive semidefinite matrix, matrix inequalities, and matrix eigenvalue inequalities, we propose a new two-parameter-type upper solution bound of the CCARE. Next, we present an iterative algorithm for finding the tighter upper solution bound of CCARE, prove its boundedness, and analyse its monotonicity and convergence. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.

scholarly journals Parameterized Nonlinear Least Squares for Unsupervised Nonlinear Spectral Unmixing

2019 ◽  
Vol 11 (2) ◽  
pp. 148 ◽  
Author(s):  
Risheng Huang ◽  
Xiaorun Li ◽  
Haiqiang Lu ◽  
Jing Li ◽  
Liaoying Zhao
Keyword(s):  
Least Squares ◽  
Optimization Problem ◽  
Nonnegative Matrix Factorization ◽  
Optimization Problems ◽  
State Of The Art ◽  
Synthetic Data ◽  
Nonnegative Matrix ◽  
Nonlinear Least Squares ◽  
Hyperspectral Data ◽  
Spectral Unmixing

This paper presents a new parameterized nonlinear least squares (PNLS) algorithm for unsupervised nonlinear spectral unmixing (UNSU). The PNLS-based algorithms transform the original optimization problem with respect to the endmembers, abundances, and nonlinearity coefficients estimation into separate alternate parameterized nonlinear least squares problems. Owing to the Sigmoid parameterization, the PNLS-based algorithms are able to thoroughly relax the additional nonnegative constraint and the nonnegative constraint in the original optimization problems, which facilitates finding a solution to the optimization problems . Subsequently, we propose to solve the PNLS problems based on the Gauss–Newton method. Compared to the existing nonnegative matrix factorization (NMF)-based algorithms for UNSU, the well-designed PNLS-based algorithms have faster convergence speed and better unmixing accuracy. To verify the performance of the proposed algorithms, the PNLS-based algorithms and other state-of-the-art algorithms are applied to synthetic data generated by the Fan model and the generalized bilinear model (GBM), as well as real hyperspectral data. The results demonstrate the superiority of the PNLS-based algorithms.

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