arXiv subjects

Dong-Hui Li

Publications and source records attributed to Dong-Hui Li.

7 recordsLinked to original sources

arXiv2025

A Variational Characterization and A Line Search Newton-Noda Method for the unifying spectral problem of nonnegative tensors

We study the general $(\boldsymbol{\sigma},\mathbf{p})$-eigenvalue problem of nonnegative tensors introduced by A. Gautier, F. Tudisco, and M. Hein [SIAM J. Matrix Anal. Appl., 40 (2019), pp. 1206--1231], which unifies several well-studied tensor eigenvalue and singular value problems. First, we propose an alternative min-max Collatz--Wielandt formula for the $(\boldsymbol{\sigma},\mathbf{p})$-spectral radius, which bypasses the auxiliary multihomogeneous mapping employed in that work. This variational characterization both recovers several classical results and admits a natural convex reformulation. It arises from an alternative approach that directly connects the $(\boldsymbol{\sigma},\mathbf{p})$-spectral problem to a class of convex programs. We then develop and analyze a line search Newton-Noda method (LS-NNM) for computing the positive $(\boldsymbol{\sigma},\mathbf{p})$-eigenpair of nonnegative tensors. The proposed method integrates Newton method with Noda iteration. The Newton equation is derived from an equivalent nonlinear system, while the eigenvalue sequence is updated by the strategy of the Noda iteration and its variants. To ensure global convergence, we introduce a positivity-preserving line search procedure based on an equivalent constrained optimization problem. The global and quadratic convergence of LS-NNM are established for the class of $(\boldsymbol{\sigma},\mathbf{p})$-spectral problem that admits a unique positive $(\boldsymbol{\sigma},\mathbf{p})$-eigenpair, as guaranteed by the Perron-Frobenius theorem. Finally, numerical experiments are conducted to illustrate the performance of LS-NNM.

math.OC↗

arXiv2024

Support matrix machine: exploring sample sparsity, low rank, and adaptive sieving in high-performance computing

Support matrix machine (SMM) is a successful supervised classification model for matrix-type samples. Unlike support vector machines, it employs low-rank regularization on the regression matrix to effectively capture the intrinsic structure embedded in each input matrix. When solving a large-scale SMM, a major challenge arises from the potential increase in sample size, leading to substantial computational and storage burdens. To address these issues, we design a semismooth Newton-CG (SNCG) based augmented Lagrangian method (ALM) for solving the SMM. The ALM exhibits an asymptotic R-superlinear convergence if a strict complementarity condition is satisfied. The SNCG method is employed to solve the ALM subproblems, achieving at least a superlinear convergence rate under the nonemptiness of an index set. Furthermore, the sparsity of samples and the low-rank nature of solutions enable us to reduce the computational cost and storage demands for the Newton linear systems. Additionally, we develop an adaptive sieving strategy that generates a solution path for the SMM by exploiting sample sparsity. The finite convergence of this strategy is also demonstrated. Numerical experiments on both large-scale real and synthetic datasets validate the effectiveness of the proposed methods.

math.OC↗

arXiv2023

A Feasible Conjugate Gradient Method for Calculating $\mathcal B$-Eigenpairs of Symmetric Tensors

In this paper, we propose a feasible conjugate gradient (FCG) method for calculating ${\mathcal B}$-eigenpairs of a symmetric tensor ${\mathcal A}$. The method is an extension of the well-known conjugate gradient method for unconstrained optimization problems to some curve constrained optimization problems. The proposed FCG method can find a ${\mathcal B}$-eigenpair of a symmetric tensor ${\mathcal A}$ without the requirement that the orders of ${\mathcal A}$ and $\mathcal B$ are equal. We pay particular attention to the Polak-Ribíre-Polyak (PRP) type conjugate gradient method. We show that the FCG method with some Armijo-type line search is globally convergent. Our numerical experiments indicate the promising performance of the proposed method.

math.OC↗

arXiv2023

Finding the spectral radius of a nonnegative irreducible symmetric tensor via DC programming

The Perron-Frobenius theorem says that the spectral radius of an irreducible nonnegative tensor is the unique positive eigenvalue corresponding to a positive eigenvector. With this in mind, the purpose of this paper is to find the spectral radius and its corresponding positive eigenvector of an irreducible nonnegative symmetric tensor. By transferring the eigenvalue problem into an equivalent problem of minimizing a concave function on a closed convex set, which is typically a DC (difference of convex functions) programming, we derive a simpler and cheaper iterative method. The proposed method is well-defined. Furthermore, we show that both sequences of the eigenvalue estimates and the eigenvector evaluations generated by the method $Q$-linearly converge to the spectral radius and its corresponding eigenvector, respectively. To accelerate the method, we introduce a line search technique. The improved method retains the same convergence property as the original version. Preliminary numerical results show that the improved method performs quite well.

math.OC↗

arXiv2020

A Lower Dimensional Linear Equation Approach to The M-Tensor Complementarity Problem

We are interested in finding a solution to the tensor complementarity problem with a strong M-tensor, which we call the M-tensor complementarity problem. We propose a lower dimensional linear equation approach to solve that problem. At each iteration, only a lower dimensional system of linear equation needs to be solved. The coefficient matrices of the lower dimensional linear systems are independent of the iteration after finitely many iterations. We show that starting from zero or some nonnegative point, the method generates a sequence of iterates that converges to a solution of the problem monotonically. We then make an improvement to the method and establish its monotone convergence. At last, we do numerical experiments to test the proposed methods. The results positively support the proposed methods.

math.OC↗

arXiv2020

Inexact Newton Method for M-Tensor Equations

We first investigate properties of M-tensor equations. In particular, we show that if the constant term of the equation is nonnegative, then finding a nonnegative solution of the equation can be done by finding a positive solution of a lower dimensional M-tensor equation. We then propose an inexact Newton method to find a positive solution to the lower dimensional equation and establish its global convergence. We also show that the convergence rate of the method is quadratic. At last, we do numerical experiments to test the proposed Newton method. The results show that the proposed Newton method has a very good numerical performance.

math.OC↗

arXiv2018

Finding a Nonnegative Solution to an M-Tensor Equation

We are concerned with the tensor equation with an M-tensor or Z-tensor, which we call the M- tensor equation or Z-tensor equation respectively. We derive a necessary and sufficient condition for a Z (or M)-tensor equation to have nonnegative solutions. We then develop a monotone iterative method to find a nonnegative solution to an M-tensor equation. The method can be regarded as an approximation to Newton's method for solving the equation. At each iteration, we solve a system of linear equations. An advantage of the proposed method is that the coefficient matrices of the linear systems are independent of the iteration. We show that if the initial point is appropriately chosen, then the sequence of iterates generated by the method converges to a nonnegative solution of the M- tensor equation monotonically and linearly. At last, we do numerical experiments to test the proposed methods. The results show the efficiency of the proposed methods.

math.OC↗