Related papers: Convergence theory for preconditioned eigenvalue s…
Preconditioned eigenvalue solvers (eigensolvers) are gaining popularity, but their convergence theory remains sparse and complex. We consider the simplest preconditioned eigensolver--the gradient iterative method with a fixed step size--for…
Different variants of approximate inverse iteration like the locally optimal block preconditioned conjugate gradient method became in recent years increasingly popular for the solution of the large matrix eigenvalue problems arising from…
Preconditioned gradient iterations for very large eigenvalue problems are efficient solvers with growing popularity. However, only for the simplest preconditioned eigensolver, namely the preconditioned gradient iteration (or preconditioned…
The extended Davis-Kahan theorem makes use of polynomial matrix transformations to produce bounds at least as tight as the standard Davis-Kahan theorem. The optimization problem of finding transformation parameters resulting in optimal…
The real-time solution of parametric optimization problems is critical for applications that demand high accuracy under tight real-time constraints, such as model predictive control. To this end, this work presents a learning-based…
It is proved that, for an indefinite quadratic programming problem under linear constraints, any iterative sequence generated by the Proximal DC decomposition algorithm $R$-linearly converges to a Karush-Kuhn-Tucker point, provided that the…
Many fields of science and engineering require finding eigenvalues and eigenvectors of large matrices. The solutions can represent oscillatory modes of a bridge, a violin, the disposition of electrons around an atom or molecule, the…
Preconditioned eigenvalue solvers offer the possibility to incorporate preconditioners for the solution of large-scale eigenvalue problems, as they arise from the discretization of partial differential equations. The convergence analysis of…
This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection-diffusion-reaction problems discretized by Galerkin or…
We propose an inexact proximal augmented Lagrangian framework with explicit inner problem termination rule for composite convex optimization problems. We consider arbitrary linearly convergent inner solver including in particular stochastic…
We provide an interior point method based on quasi-Newton iterations, which only requires first-order access to a strongly self-concordant barrier function. To achieve this, we extend the techniques of Dunagan-Harvey [STOC '07] to maintain…
The performance of optimization methods is often tied to the spectrum of the objective Hessian. Yet, conventional assumptions, such as smoothness, do often not enable us to make finely-grained convergence statements -- particularly not for…
A new iterative method for solving large scale symmetric nonlinear eigenvalue problems is presented. We firstly derive an infinite dimensional symmetric linearization of the nonlinear eigenvalue problem, then we apply the indefinite Lanczos…
Several problems in machine learning, statistics, and other fields rely on computing eigenvectors. For large scale problems, the computation of these eigenvectors is typically performed via iterative schemes such as subspace iteration or…
This paper establishes new upper bounds for the sum of the $k$ largest eigenvalues of symmetric matrices. When applied to the adjacency matrix of a graph, our results improve upon a related bound due to Mohar {\bf [On the sum of k largest…
This paper investigates using the conjugate gradient iterative solver for ill-posed problems. We show that preconditioner and Tikhonov-regularization work in conjunction. In particular when they employ the same symmetric positive…
We study a continuous-time dynamical system which arises as the limit of a broad class of nonlinearly preconditioned gradient methods. Under mild assumptions, we establish existence of global solutions and derive Lyapunov-based convergence…
In this note we present a parameterized class of lower triangular matrices. The components of the eigenvectors grow rapidly and will exceed the representational range of any finite number system. The eigenvalues and the eigenvectors are…
Dual first-order methods are powerful techniques for large-scale convex optimization. Although an extensive research effort has been devoted to studying their convergence properties, explicit convergence rates for the primal iterates have…
This paper develops the preconditioning technique as a method to address the accuracy issue caused by ill-conditioning. Given a preconditioner $M$ for an ill-conditioned linear system $Ax=b$, we show that, if the inverse of the…